r"""
Ambient spaces of modular symbols
This module defines the following classes. There is an abstract base
class ``ModularSymbolsAmbient``, derived from
``space.ModularSymbolsSpace`` and ``hecke.AmbientHeckeModule``. As
this is an abstract base class, only derived classes should be
instantiated. There are five derived classes:
- ``ModularSymbolsAmbient_wtk_g0``, for modular symbols of general
weight `k` for `\Gamma_0(N)`;
- ``ModularSymbolsAmbient_wt2_g0`` (derived from
``ModularSymbolsAmbient_wtk_g0``), for modular symbols of weight 2
for `\Gamma_0(N)`;
- ``ModularSymbolsAmbient_wtk_g1``, for modular symbols of general
weight `k` for `\Gamma_1(N)`;
- ``ModularSymbolsAmbient_wtk_gamma_h``, for modular symbols of
general weight `k` for `\Gamma_H`, where `H` is a subgroup of
`\ZZ/N\ZZ`;
- ``ModularSymbolsAmbient_wtk_eps``, for modular symbols of general
weight `k` and character `\epsilon`.
EXAMPLES:
We compute a space of modular symbols modulo 2. The dimension is
different from that of the corresponding space in characteristic
0::
sage: M = ModularSymbols(11,4,base_ring=GF(2)); M
Modular Symbols space of dimension 7 for Gamma_0(11) of weight 4
with sign 0 over Finite Field of size 2
sage: M.basis()
([X*Y,(1,0)], [X*Y,(1,8)], [X*Y,(1,9)], [X^2,(0,1)], [X^2,(1,8)], [X^2,(1,9)], [X^2,(1,10)])
sage: M0 = ModularSymbols(11,4,base_ring=QQ); M0
Modular Symbols space of dimension 6 for Gamma_0(11) of weight 4
with sign 0 over Rational Field
sage: M0.basis()
([X^2,(0,1)], [X^2,(1,6)], [X^2,(1,7)], [X^2,(1,8)], [X^2,(1,9)], [X^2,(1,10)])
The characteristic polynomial of the Hecke operator `T_2` has an extra
factor `x`.
::
sage: M.T(2).matrix().fcp('x')
(x + 1)^2 * x^5
sage: M0.T(2).matrix().fcp('x')
(x - 9)^2 * (x^2 - 2*x - 2)^2
"""
from sage.misc.search import search
import sage.misc.latex as latex
import sage.misc.misc as misc
import sage.matrix.matrix_space as matrix_space
import sage.modules.free_module_element as free_module_element
import sage.modules.free_module as free_module
import sage.misc.misc as misc
import sage.modular.arithgroup.all as arithgroup
import sage.modular.dirichlet as dirichlet
import sage.modular.hecke.all as hecke
import sage.rings.rational_field as rational_field
import sage.rings.integer_ring as integer_ring
import sage.rings.all as rings
import sage.rings.arith as arith
import sage.structure.formal_sum as formal_sum
import sage.categories.all as cat
from sage.modular.cusps import Cusp
import sage.structure.all
import boundary
import element
import heilbronn
import manin_symbols
import modular_symbols
import modsym
import p1list
import relation_matrix
import space
import subspace
QQ = rings.Rational
ZZ = rings.Integers
class ModularSymbolsAmbient(space.ModularSymbolsSpace, hecke.AmbientHeckeModule):
r"""
An ambient space of modular symbols for a congruence subgroup of
`SL_2(\ZZ)`.
This class is an abstract base class, so only derived classes
should be instantiated.
INPUT:
- ``weight`` - an integer
- ``group`` - a congruence subgroup.
- ``sign`` - an integer, either -1, 0, or 1
- ``base_ring`` - a commutative ring
- ``custom_init`` - a function that is called with self as input
before any computations are done using self; this could be used
to set a custom modular symbols presentation.
"""
def __init__(self, group, weight, sign, base_ring,
character=None, custom_init=None):
"""
Initialize a space of modular symbols.
INPUT:
- ``weight`` - an integer
- ``group`` - a congruence subgroup.
- ``sign`` - an integer, either -1, 0, or 1
- ``base_ring`` - a commutative ring
EXAMPLES::
sage: ModularSymbols(2,2)
Modular Symbols space of dimension 1 for Gamma_0(2) of weight 2 with sign 0 over Rational Field
"""
weight = int(weight)
if weight <= 1:
raise ValueError, "Weight (=%s) Modular symbols of weight <= 1 not defined."%weight
if not arithgroup.is_CongruenceSubgroup(group):
raise TypeError, "group must be a congruence subgroup"
sign = int(sign)
if not isinstance(base_ring, rings.Ring) and base_ring.is_field():
raise TypeError, "base_ring must be a commutative ring"
if character == None and arithgroup.is_Gamma0(group):
character = dirichlet.TrivialCharacter(group.level(), base_ring)
space.ModularSymbolsSpace.__init__(self, group, weight,
character, sign, base_ring)
if custom_init is not None:
custom_init(self)
try:
formula = self._dimension_formula()
except NotImplementedError:
formula = None
rank = self.rank()
if formula != None:
assert rank == formula, \
"Computed dimension (=%s) of ambient space \"%s\" doesn't match dimension formula (=%s)!\n"%(d, self, formula) + \
"ModularSymbolsAmbient: group = %s, weight = %s, sign = %s, base_ring = %s, character = %s"%(
group, weight, sign, base_ring, character)
hecke.AmbientHeckeModule.__init__(self, base_ring, rank, group.level(), weight)
def __cmp__(self, other):
"""
Standard comparison function.
EXAMPLES::
sage: ModularSymbols(11,2) == ModularSymbols(11,2) # indirect doctest
True
sage: ModularSymbols(11,2) == ModularSymbols(11,4) # indirect doctest
False
"""
if not isinstance(other, space.ModularSymbolsSpace):
return cmp(type(self), type(other))
if isinstance(other, ModularSymbolsAmbient):
return misc.cmp_props(self, other, ['group', 'weight', 'sign', 'base_ring', 'character'])
c = cmp(self, other.ambient_hecke_module())
if c: return c
if self.free_module() == other.free_module():
return 0
return -1
def new_submodule(self, p=None):
r"""
Returns the new or `p`-new submodule of this modular symbols ambient space.
INPUT:
- ``p`` - (default: None); if not None, return only
the `p`-new submodule.
OUTPUT:
The new or `p`-new submodule of this modular symbols ambient space.
EXAMPLES::
sage: ModularSymbols(100).new_submodule()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 31 for Gamma_0(100) of weight 2 with sign 0 over Rational Field
sage: ModularSymbols(389).new_submodule()
Modular Symbols space of dimension 65 for Gamma_0(389) of weight 2 with sign 0 over Rational Field
"""
if self.level().is_prime() and self.weight() == 2:
return self
return hecke.AmbientHeckeModule.new_submodule(self, p=p)
def manin_symbols(self):
"""
Return the list of Manin symbols for this modular symbols ambient space.
EXAMPLES::
sage: ModularSymbols(11,2).manin_symbols()
Manin Symbol List of weight 2 for Gamma0(11)
"""
raise NotImplementedError
def manin_generators(self):
"""
Return list of all Manin symbols for this space. These are the
generators in the presentation of this space by Manin symbols.
EXAMPLES::
sage: M = ModularSymbols(2,2)
sage: M.manin_generators()
[(0,1), (1,0), (1,1)]
::
sage: M = ModularSymbols(1,6)
sage: M.manin_generators()
[[Y^4,(0,0)], [X*Y^3,(0,0)], [X^2*Y^2,(0,0)], [X^3*Y,(0,0)], [X^4,(0,0)]]
"""
return self._manin_generators
def manin_basis(self):
r"""
Return a list of indices into the list of Manin generators (see
``self.manin_generators()``) such that those symbols
form a basis for the quotient of the `\QQ`-vector
space spanned by Manin symbols modulo the relations.
EXAMPLES::
sage: M = ModularSymbols(2,2)
sage: M.manin_basis()
[1]
sage: [M.manin_generators()[i] for i in M.manin_basis()]
[(1,0)]
sage: M = ModularSymbols(6,2)
sage: M.manin_basis()
[1, 10, 11]
sage: [M.manin_generators()[i] for i in M.manin_basis()]
[(1,0), (3,1), (3,2)]
"""
try:
return self._manin_basis
except AttributeError:
self.compute_presentation()
return self._manin_basis
def p1list(self):
"""
Return a P1list of the level of this modular symbol space.
EXAMPLES::
sage: ModularSymbols(11,2).p1list()
The projective line over the integers modulo 11
"""
try:
return self.__p1list
except AttributeError:
self.__p1list = p1list.P1List(self.level())
return self.__p1list
def compute_presentation(self):
r"""
Compute and cache the presentation of this space.
EXAMPLES::
sage: ModularSymbols(11,2).compute_presentation() # no output
"""
B, basis, mod = relation_matrix.compute_presentation(
self.manin_symbols(), self.sign(),
self.base_ring())
self._manin_generators = self.manin_symbols().manin_symbol_list()
self._manin_basis = basis
self._manin_gens_to_basis = B
self._mod2term = mod
def manin_gens_to_basis(self):
r"""
Return the matrix expressing the manin symbol generators in terms of the basis.
EXAMPLES::
sage: ModularSymbols(11,2).manin_gens_to_basis()
[-1 0 0]
[ 1 0 0]
[ 0 0 0]
[ 0 0 1]
[ 0 -1 1]
[ 0 -1 0]
[ 0 0 -1]
[ 0 0 -1]
[ 0 1 -1]
[ 0 1 0]
[ 0 0 1]
[ 0 0 0]
"""
try:
return self._manin_gens_to_basis
except AttributeError:
self.compute_presentation()
return self._manin_gens_to_basis
def __call__(self, x, computed_with_hecke=False):
r"""
Coerce `x` into this modular symbols space. The result is
either an element of self or a subspace of self.
INPUTS:
The allowed input types for `x` are as follows:
- ``Vector`` - a vector of the same degree. This
defines the corresponding linear combination of the basis of self.
- ``ManinSymbol`` - a Manin symbol of the same weight
as the space
- ``ModularSymbolsElement`` - a modular symbol whose
ambient parent is this space of modular symbols. (TODO: make more
sophisticated)
- 0 - the integer 0; results in the 0 modular symbol.
- 3-tuple - Given a 3-tuple (i,u,v), returns the modular symbol
element defined by the Manin symbol
`[X^{i}\cdot Y^{k-2-i}, (u,v)]`, where k is the weight.
Note that we must have `0\leq i \leq k-2`.
- 2-tuple - Given a 2-tuple (u,v), returns the element defined by
the Manin symbol `[X^0 \cdot Y^{2-k}, (u,v)]`.
- 2-elements list - Given a list ``[alpha, beta]``,
where `\alpha` and `\beta` are (coercible to)
cusps, return the modular symbol `\{\alpha, \beta\}`. When
the the weight `k > 2` return
`Y^{k-2} \{\alpha, \beta\}`.
- 3-element list - Given a list ``[i, alpha, beta]``,
where `i` is an integer, and `\alpha`,
`\beta` are (coercible to) cusps, return the modular symbol
`X^i Y^{k-2-i} \{\alpha, \beta\}`.
If our list is ``[f, alpha, beta]``, where `f`
is a homogeneous polynomial in two variables of degree k-2 with
integer coefficients, and alpha and beta are cusps, return the
corresponding sum of modular symbols as an element of self. So if
`f = \sum_{i=0}^{k-2} a_i X^i Y^{k-2-i}`, return
`\sum_{i=0}^{k-2} a_i * [ i, alpha, beta ]`.
EXAMPLES::
sage: M = ModularSymbols(37,2)
M(0) is the 0 element of the space::
sage: M(0)
0
sage: type(M(0))
<class 'sage.modular.modsym.element.ModularSymbolsElement'>
From a vector of the correct dimension we construct the
corresponding linear combination of the basis elements::
sage: M.dimension()
5
sage: M.basis()
((1,0), (1,23), (1,32), (1,34), (1,35))
sage: M(vector([1,2,3,4,5]))
(1,0) + 2*(1,23) + 3*(1,32) + 4*(1,34) + 5*(1,35)
sage: M(vector([1/2,2/3,3/4,4/5,5/6]))
1/2*(1,0) + 2/3*(1,23) + 3/4*(1,32) + 4/5*(1,34) + 5/6*(1,35)
Manin symbols can be converted to elements of the space::
sage: from sage.modular.modsym.manin_symbols import ManinSymbol
sage: ManinSymbol(M.manin_symbols(),(0,2,3))
(2,3)
sage: M(ManinSymbol(M.manin_symbols(),(0,2,3)))
(1,34) - (1,35)
However, it is easier to use one of the following forms.
Either a 3-tuple `(i,u,v)` or a 2-tuple `(u,v)` with `i=0`
assumed::
sage: M((0,2,3))
(1,34) - (1,35)
sage: M((2,3))
(1,34) - (1,35)
Or a 3-list `[i,\alpha,\beta]` where `i` is the degree and
`\alpha` and `beta` are cusps, or a 2-tuple `[\alpha,\beta]`
with `i=0` assumed::
sage: M([0,Cusp(1/2),Cusp(0)])
(1,35)
sage: M([Cusp(1/2),Cusp(0)])
(1,35)
"""
if isinstance(x, free_module_element.FreeModuleElement):
if x.degree() != self.dimension():
raise TypeError, "Incompatible degrees: x has degree %s but modular symbols space has dimension %s"%(
x.degree(), self.dimension())
return element.ModularSymbolsElement(self, x)
elif isinstance(x, (manin_symbols.ManinSymbol, element.ModularSymbolsElement)):
return self.element(x)
elif isinstance(x, modular_symbols.ModularSymbol):
return self(x.manin_symbol_rep())
elif isinstance(x, (int, rings.Integer)) and x==0:
return element.ModularSymbolsElement(self, self.free_module()(0))
elif isinstance(x, tuple):
return self.manin_symbol(x)
elif isinstance(x, formal_sum.FormalSum):
return sum([c*self(y) for c, y in x], self(0))
elif isinstance(x, list):
if len(x) == 3 and rings.is_MPolynomial(x[0]):
return self.modular_symbol_sum(x)
else:
return self.modular_symbol(x)
raise TypeError, "No coercion of %s into %s defined."%(x, self)
def change_ring(self, R):
r"""
Change the base ring to R.
EXAMPLES::
sage: ModularSymbols(Gamma1(13), 2).change_ring(GF(17))
Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Finite Field of size 17
sage: M = ModularSymbols(DirichletGroup(5).0, 7); MM=M.change_ring(CyclotomicField(8)); MM
Modular Symbols space of dimension 6 and level 5, weight 7, character [zeta8^2], sign 0, over Cyclotomic Field of order 8 and degree 4
sage: MM.change_ring(CyclotomicField(4)) == M
True
sage: M.change_ring(QQ)
Traceback (most recent call last):
...
ValueError: cannot coerce element of order 4 into self
"""
if self.character() is None:
return modsym.ModularSymbols(self.group(), self.weight(), self.sign(), R)
else:
return modsym.ModularSymbols(self.character(), self.weight(), self.sign(), R)
def base_extend(self, R):
r"""
Canonically change the base ring to R.
EXAMPLE::
sage: M = ModularSymbols(DirichletGroup(5).0, 7); MM = M.base_extend(CyclotomicField(8)); MM
Modular Symbols space of dimension 6 and level 5, weight 7, character [zeta8^2], sign 0, over Cyclotomic Field of order 8 and degree 4
sage: MM.base_extend(CyclotomicField(4))
Traceback (most recent call last):
...
ValueError: No coercion defined
"""
if not R.has_coerce_map_from(self.base_ring()):
raise ValueError, "No coercion defined"
else:
return self.change_ring(R)
def _action_on_modular_symbols(self, g):
r"""
Returns the matrix of the action of a 2x2 matrix on this space.
INPUT:
``g`` (list) -- `g=[a,b,c,d]` where `a,b,c,d` are integers
defining a `2\times2` integer matrix.
OUTPUT:
(matrix) The matrix of the action of `g` on this Modular
Symbol space, with respect to the standard basis.
.. note::
Use _matrix_of_operator_on_modular_symbols for more general
operators.
EXAMPLES::
sage: M = ModularSymbols(11,4,1)
sage: M._action_on_modular_symbols([1,2,3,7])
[ 0 0 5/2 -3/2]
[ 0 0 5/2 -3/2]
[ 0 1 0 0]
[ 0 1 -1/2 1/2]
"""
if not isinstance(g, list):
raise TypeError, "g must be a list"
if not len(g) == 4:
raise TypeError, "g must be a list of length 4"
return self._matrix_of_operator_on_modular_symbols(self, [g])
def manin_symbol(self, x, check=True):
r"""
Construct a Manin Symbol from the given data.
INPUT:
- ``x`` (list) -- either `[u,v]` or `[i,u,v]`, where `0\le
i\le k-2` where `k` is the weight, and `u`,`v` are integers
defining a valid element of `\mathbb{P}^1(N)`, where `N` is
the level.
OUTPUT:
(ManinSymbol) the monomial Manin Symbol associated to
`[i;(u,v)]`, with `i=0` if not supplied, corresponding to the
symbol `[X^i*Y^{k-2-i}, (u,v)]`.
EXAMPLES::
sage: M = ModularSymbols(11,4,1)
sage: M.manin_symbol([2,5,6])
[X^2,(1,10)]
"""
if check:
if len(x) == 2:
x = (0,x[0],x[1])
if len(x) == 3:
if x[0] < 0 or x[0] > self.weight()-2:
raise ValueError, "The first entry of the tuple (=%s) must be an integer between 0 and k-2 (=%s)."%(
x, self.weight()-2)
else:
raise ValueError, "x (=%s) must be of length 2 or 3"%x
N = self.level()
x = (x[0], x[1]%N, x[2]%N)
try:
return self.__manin_symbol[x]
except AttributeError:
self.__manin_symbol = {}
except KeyError:
pass
y = manin_symbols.ManinSymbol(self.manin_symbols(), x)
z = self(y)
self.__manin_symbol[x] = z
return z
def _modular_symbol_0_to_alpha(self, alpha, i=0):
r"""
Return the modular symbol `\{0,\alpha\}` in this space.
INPUT:
- ``alpha`` (rational or Infinity) -- a cusp
- ``i`` (int, default 0) -- the degree of the symbol.
OUTPUT:
(ModularSymbol) The modular symbol `X^iY^{k-2-i}\{0,\alpha\}`.
EXAMPLES::
sage: M = ModularSymbols(11,4,1)
sage: M._modular_symbol_0_to_alpha(Cusp(3/5))
11*[X^2,(1,7)] + 33*[X^2,(1,9)] - 20*[X^2,(1,10)]
sage: M._modular_symbol_0_to_alpha(Cusp(3/5),1)
15/2*[X^2,(1,7)] + 35/2*[X^2,(1,9)] - 10*[X^2,(1,10)]
sage: M._modular_symbol_0_to_alpha(Cusp(Infinity))
-[X^2,(1,10)]
sage: M._modular_symbol_0_to_alpha(Cusp(Infinity),1)
0
"""
if alpha.is_infinity():
return self.manin_symbol((i,0,1), check=False)
v, c = arith.continued_fraction_list(alpha._rational_(), partial_convergents=True)
a = self(0)
zero = rings.ZZ(0)
one = rings.ZZ(1)
two = rings.ZZ(2)
if self.weight() > two:
R = rings.ZZ['X']
X = R.gen(0)
a += self.manin_symbol((i,0,1), check=False)
for k in range(0,len(c)):
if k == 0:
x = c[0][0]
y = -1
z = 1
w = 0
else:
x = c[k][0]
y = c[k-1][0]
z = c[k][1]
w = c[k-1][1]
if k%2 == 0:
y = -y
w = -w
p1 = x*X+y
p2 = z*X+w
if i == 0:
p1 = R(one)
if (self.weight()-2-i == 0):
p2 = R(one)
poly = (p1**i) * (p2**(self.weight()-2-i))
for s in range(0,self.weight()-1):
a += poly[s] * self.manin_symbol((s,z,w), check=False)
else:
for k in range(1,len(c)):
u = c[k][1]
v = c[k-1][1]
if k % 2 == 0:
v = -v
x = self.manin_symbol((i, u, v), check=False)
a += x
return a
def modular_symbol(self, x, check=True):
r"""
Create a modular symbol in this space.
INPUT:
- ``x`` (list) -- a list of either 2 or 3 entries:
- 2 entries: `[\alpha, \beta]` where `\alpha` and `\beta`
are cusps;
- 3 entries: `[i, \alpha, \beta]` where `0\le i\le k-2`
and `\alpha` and `\beta` are cusps;
- ``check`` (bool, default True) -- flag that determines
whether the input ``x`` needs processing: use check=False
for efficiency if the input ``x`` is a list of length 3 whose
first entry is an Integer, and whose second and third
entries are Cusps (see examples).
OUTPUT:
(Modular Symbol) The modular symbol `Y^{k-2}\{\alpha,
\beta\}`. or `X^i Y^{k-2-i}\{\alpha,\beta\}`.
EXAMPLES::
sage: set_modsym_print_mode('modular')
sage: M = ModularSymbols(11)
sage: M.modular_symbol([2/11, oo])
-{-1/9, 0}
sage: M.1
{-1/8, 0}
sage: M.modular_symbol([-1/8, 0])
{-1/8, 0}
sage: M.modular_symbol([0, -1/8, 0])
{-1/8, 0}
sage: M.modular_symbol([10, -1/8, 0])
Traceback (most recent call last):
...
ValueError: The first entry of the tuple (=[10, -1/8, 0]) must be an integer between 0 and k-2 (=0).
::
sage: N = ModularSymbols(6,4)
sage: set_modsym_print_mode('manin')
sage: N([1,Cusp(-1/4),Cusp(0)])
17/2*[X^2,(2,3)] - 9/2*[X^2,(2,5)] + 15/2*[X^2,(3,1)] - 15/2*[X^2,(3,2)]
sage: N([1,Cusp(-1/2),Cusp(0)])
1/2*[X^2,(2,3)] + 3/2*[X^2,(2,5)] + 3/2*[X^2,(3,1)] - 3/2*[X^2,(3,2)]
Use check=False for efficiency if the input x is a list of length 3
whose first entry is an Integer, and whose second and third entries
are cusps::
sage: M.modular_symbol([0, Cusp(2/11), Cusp(oo)], check=False)
-(1,9)
::
sage: set_modsym_print_mode() # return to default.
"""
if check:
if len(x) == 2:
x = [0,x[0],x[1]]
elif len(x) == 3:
if x[0] < 0 or x[0] > self.weight()-2:
raise ValueError, "The first entry of the tuple (=%s) must be an integer between 0 and k-2 (=%s)."%(
x, self.weight()-2)
else:
raise ValueError, "x (=%s) must be of length 2 or 3"%x
i = rings.Integer(x[0])
alpha = Cusp(x[1])
beta = Cusp(x[2])
else:
i = x[0]
alpha = x[1]
beta = x[2]
a = self._modular_symbol_0_to_alpha(alpha, i)
b = self._modular_symbol_0_to_alpha(beta, i)
return b - a
def modular_symbol_sum(self, x, check=True):
r"""
Construct a modular symbol sum.
INPUT:
- ``x`` (list) -- `[f, \alpha, \beta]` where `f =
\sum_{i=0}^{k-2} a_i X^i Y^{k-2-i}` is a homogeneous
polynomial over `\ZZ` of degree `k` and `\alpha` and `\beta`
are cusps.
- ``check`` (bool, default True) -- if True check the validity
of the input tuple ``x``
OUTPUT:
The sum `\sum_{i=0}^{k-2} a_i [ i, \alpha, \beta ]` as an
element of this modular symbol space.
EXAMPLES:
sage: M = ModularSymbols(11,4)
sage: R.<X,Y>=QQ[]
sage: M.modular_symbol_sum([X*Y,Cusp(0),Cusp(Infinity)])
-3/14*[X^2,(1,6)] + 1/14*[X^2,(1,7)] - 1/14*[X^2,(1,8)] + 1/2*[X^2,(1,9)] - 2/7*[X^2,(1,10)]
"""
if check:
if len(x) != 3:
raise ValueError, "%s must have length 3"%x
f = x[0]
R = self.base_ring()['X','Y']
X = R.gen(0)
try:
f = R(f)
except TypeError:
raise ValueError, \
"f must be coercible to a polynomial over %s"%self.base_ring()
if (not f.is_homogeneous()) or (f.degree() != self.weight()-2):
raise ValueError, "f must be a homogeneous polynomial of degree k-2"
alpha = Cusp(x[1])
beta = Cusp(x[2])
else:
f = x[0]
R = self.base_ring()
X = R.gen(0)
alpha = x[1]
beta = x[2]
s = self(0)
for term in f.monomials():
deg = term.degree(X)
a = self._modular_symbol_0_to_alpha(alpha, deg)
b = self._modular_symbol_0_to_alpha(beta, deg)
s += f.monomial_coefficient(term) * (b-a)
return s
def _compute_dual_hecke_matrix(self, n):
r"""
Return the matrix of the dual Hecke operator `T(n)`.
INPUT:
- ``n`` (int) -- a positive integer
OUTPUT:
(matrix) The matrix of the dual od `T(n)`.
EXAMPLES::
sage: M = ModularSymbols(11,4,1)
sage: M._compute_dual_hecke_matrix(5)
[126 0 0 0]
[ 2 63 38 22]
[ 11 33 82 121]
[-13 30 6 -17]
"""
return self.hecke_matrix(n).transpose()
def _compute_hecke_matrix_prime(self, p, rows=None):
"""
Return the matrix of the Hecke operator `T(p)`.
INPUT:
- ``p`` (int) -- a prime number.
- ``rows`` (list or None (default)) -- if not None, a list of
the rows which should be computed; otherwise the complete
matrix will be computed,
.. note::
`p` does not have to be, prime despite the function name.
OUTPUT:
(matrix) The matrix of the Hecke operator `T(p)` on this
space, with respect to its standard basis.
ALGORITHM:
Uses Heilbronn-Cremonma matrices of `p` is prime, else use
Heilbronn-Merel matrices.
EXAMPLES:
We first compute some examples for Gamma0(N)::
sage: m = ModularSymbols(2, weight=4)
sage: m._compute_hecke_matrix_prime(2).charpoly('x')
x^2 - 9*x + 8
::
sage: m = ModularSymbols(1,weight=12)
sage: m._compute_hecke_matrix_prime(2).charpoly('x')
x^3 - 2001*x^2 - 97776*x - 1180224
sage: m._compute_hecke_matrix_prime(13).charpoly('x')
x^3 - 1792159238562*x^2 - 2070797989680255444*x - 598189440899986203208472
::
sage: m = ModularSymbols(1,weight=12, sign=-1)
sage: m._compute_hecke_matrix_prime(5)
[4830]
sage: m._compute_hecke_matrix_prime(23)
[18643272]
::
sage: m = ModularSymbols(3,4)
sage: m._compute_hecke_matrix_prime(2).charpoly('x')
x^2 - 18*x + 81
::
sage: m = ModularSymbols(6,4)
sage: m._compute_hecke_matrix_prime(2).charpoly('x')
x^6 - 14*x^5 + 29*x^4 + 172*x^3 - 124*x^2 - 320*x + 256
sage: m._compute_hecke_matrix_prime(3).charpoly('x')
x^6 - 50*x^5 + 511*x^4 + 3012*x^3 - 801*x^2 - 9234*x + 6561
::
sage: m = ModularSymbols(15,4, sign=-1)
sage: m._compute_hecke_matrix_prime(3).charpoly('x')
x^4 - 2*x^3 + 18*x^2 + 18*x - 243
::
sage: m = ModularSymbols(6,4)
sage: m._compute_hecke_matrix_prime(7).charpoly('x')
x^6 - 1344*x^5 + 666240*x^4 - 140462080*x^3 + 8974602240*x^2 + 406424518656*x + 3584872677376
::
sage: m = ModularSymbols(4,4)
sage: m._compute_hecke_matrix_prime(3).charpoly('x')
x^3 - 84*x^2 + 2352*x - 21952
We now compute some examples for modular symbols on Gamma1(N)::
sage: m = ModularSymbols(Gamma1(13),2, sign=-1)
sage: m._compute_hecke_matrix_prime(2).charpoly('x')
x^2 + 3*x + 3
The following is an example with odd weight::
sage: m = ModularSymbols(Gamma1(5),3)
sage: m._compute_hecke_matrix_prime(2).charpoly('x')
x^4 - 10*x^3 + 50*x^2 - 170*x + 289
This example has composite conductor and weight2 dividing the
conductor and nontrivial sign::
sage: m = ModularSymbols(Gamma1(9),3, sign=1)
sage: m._compute_hecke_matrix_prime(3).charpoly('x')
x^6 + 3*x^4 - 19*x^3 + 24*x^2 - 9*x
In some situations we do not need all the rows of the result, and can thereby save time:
sage: m = ModularSymbols(1,weight=12)
sage: m._compute_hecke_matrix_prime(2)
[ -24 0 0]
[ 0 -24 0]
[4860 0 2049]
sage: m._compute_hecke_matrix_prime(2,rows=[0,1])
[-24 0 0]
[ 0 -24 0]
sage: m._compute_hecke_matrix_prime(2,rows=[1,2])
[ 0 -24 0]
[4860 0 2049]
"""
p = int(rings.Integer(p))
if isinstance(rows, list):
rows = tuple(rows)
try:
return self._hecke_matrices[(p,rows)]
except AttributeError:
self._hecke_matrices = {}
except KeyError:
pass
tm = misc.verbose("Computing Hecke operator T_%s"%p)
if arith.is_prime(p):
H = heilbronn.HeilbronnCremona(p)
else:
H = heilbronn.HeilbronnMerel(p)
B = self.manin_basis()
if not rows is None:
B = [B[i] for i in rows]
cols = []
mod2term = self._mod2term
R = self.manin_gens_to_basis()
K = self.base_ring()
W = R.new_matrix(nrows=len(B), ncols = R.nrows())
syms = self.manin_symbols()
n = len(syms)
j = 0
for i in B:
for h in H:
entries = syms.apply(i,h)
for k, x in entries:
f, s = mod2term[k]
if s != 0:
W[j,f] = W[j,f] + s*K(x)
j += 1
tm = misc.verbose("start matrix multiply",tm)
if hasattr(W, '_matrix_times_matrix_dense'):
Tp = W._matrix_times_matrix_dense(R)
misc.verbose("done matrix multiply and computing Hecke operator",tm)
else:
Tp = W * R
tm = misc.verbose("done matrix multiply",tm)
Tp = Tp.dense_matrix()
misc.verbose("done making Hecke operator matrix dense",tm)
self._hecke_matrices[(p,rows)] = Tp
return Tp
def __heilbronn_operator(self, M, H, t=1):
r"""
Return the matrix function to the space `M` defined by `H`, `t`.
.. note::
Users will instead use the simpler interface defined, for
example, by ``hecke_matrix()`` (see examples).
INPUT:
- ``M`` (ModularSymbols) -- codomain (a space of modular
symbols);
- ``H`` (list) -- a list of matrices in `M_2(\ZZ)`;
- ``t`` (int, default 1) -- an integer.
OUTPUT:
(free module morphism) A function from the Modular Symbol
space to the Modular Symbol space `M` defined by `t` and the
matrices in `H`.
EXAMPLES::
sage: M = ModularSymbols(37,2)
sage: M._ModularSymbolsAmbient__heilbronn_operator(M,HeilbronnCremona(3))
Hecke module morphism Heilbronn operator(The Cremona-Heilbronn matrices of determinant 3,1) defined by the matrix
[ 4 0 0 0 -1]
[ 0 -1 2 2 -2]
[ 0 2 -1 2 0]
[ 0 0 0 -3 2]
[ 0 0 0 0 1]
Domain: Modular Symbols space of dimension 5 for Gamma_0(37) of weight ...
Codomain: Modular Symbols space of dimension 5 for Gamma_0(37) of weight ...
sage: M.hecke_matrix(3)
[ 4 0 0 0 -1]
[ 0 -1 2 2 -2]
[ 0 2 -1 2 0]
[ 0 0 0 -3 2]
[ 0 0 0 0 1]
"""
MS = matrix_space.MatrixSpace(self.base_ring(), self.dimension(), M.dimension())
hom = self.Hom(M)
if self.dimension() == 0 or M.dimension() == 0:
A = MS(0)
phi = hom(A, "Heilbronn operator(%s,%s)"%(H,t))
return phi
rows = []
B = self.manin_basis()
syms = self.manin_symbols()
k = self.weight()
for n in B:
z = M(0)
i, u, v = syms[n]
for h in H:
(a,b,c,d) = tuple(h)
P = manin_symbols.apply_to_monomial(i, k-2, a,b,c,d)
(uu,vv) = (u*a+v*c, u*b+v*d)
if t != 1:
if uu%t != 0 or vv%t != 0:
continue
uu = uu//t
vv = vv//t
for m in range(len(P)):
x = M((m,uu,vv))
z += x*P[m]
rows.append(z.element())
A = MS(rows)
return hom(A, "Heilbronn operator(%s,%s)"%(H,t))
def _repr_(self):
r"""
String representation of this Modular Symbols space.
EXAMPLES::
sage: m = ModularSymbols(1,weight=12)
sage: m # indirect doctest
Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
"""
return "Modular Symbols space of dimension %s and weight %s for %s with sign %s and character %s over %s"%(
self.dimension(), self.weight(), self.group(), self.sign(), self.character()._repr_short_(), self.base_ring())
def _latex_(self):
r"""
Latex representation of this Modular Symbols space.
EXAMPLES::
sage: m = ModularSymbols(11,weight=12)
sage: latex(m) # indirect doctest
\mathrm{ModSym}_{12}(\Gamma_0(11),\left[1\right];\Bold{Q})
sage: chi = DirichletGroup(7).0
sage: m = ModularSymbols(chi)
sage: latex(m)
\mathrm{ModSym}_{2}(\Gamma_1(7),\left[\zeta_{6}\right];\Bold{Q}(\zeta_{6}))
"""
return "\\mathrm{ModSym}_{%s}(%s,%s;%s)"%(self.weight(),
latex.latex(self.group()),
latex.latex(list(self.character().values_on_gens())),
latex.latex(self.base_ring()))
def _matrix_of_operator_on_modular_symbols(self, codomain, R):
r"""
Returns the matrix of a modular symbols operator.
.. note::
Users will usually instead use the simpler interface
defined, for example, by ``hecke_matrix()`` (see examples),
though this function allows one to compute much more
general operators.
INPUT:
- ``codomain`` - space of modular symbols
- ``R`` (list) -- a list of lists `[a,b,c,d]` of length 4,
which we view as elements of `GL_2(`QQ)`.
OUTPUT:
-- (matrix) The matrix of the operator
.. math::
x \mapsto \sum_{g in R} g.x,
where `g.x` is the formal linear fractional transformation on modular
symbols, with respect to the standard basis.
EXAMPLES::
sage: M = ModularSymbols(37,2)
sage: M._matrix_of_operator_on_modular_symbols(M,HeilbronnCremona(3))
[ 4 0 0 0 0]
[ 0 -3 1 1 0]
[ 0 3 0 5 -2]
[ 0 -3 1 -5 3]
[ 0 0 2 3 -3]
"""
rows = []
for b in self.basis():
v = formal_sum.FormalSum(0, check=False)
for c, x in b.modular_symbol_rep():
for g in R:
y = x.apply(g)
v += y*c
w = codomain(v).element()
rows.append(w)
M = matrix_space.MatrixSpace(self.base_ring(), len(rows), codomain.degree(), sparse=False)
return M(rows)
def _compute_atkin_lehner_matrix(self, d):
r"""
Return the matrix of the Atkin-Lehner involution `W_d`.
INPUT:
- ``d`` (int) -- an integer that divides the level.
OUTPUT:
(matrix) The matrix of the involution `W_d` with respect to
the standard basis.
EXAMPLES: An example at level 29::
sage: M = ModularSymbols((DirichletGroup(29,QQ).0), 2,1); M
Modular Symbols space of dimension 4 and level 29, weight 2, character [-1], sign 1, over Rational Field
sage: w = M._compute_atkin_lehner_matrix(29)
sage: w^2 == 1
True
sage: w.fcp()
(x - 1)^2 * (x + 1)^2
This doesn't work since the character has order 2::
sage: M = ModularSymbols((DirichletGroup(13).0), 2,1); M
Modular Symbols space of dimension 0 and level 13, weight 2, character [zeta12], sign 1, over Cyclotomic Field of order 12 and degree 4
sage: M._compute_atkin_lehner_matrix(13)
Traceback (most recent call last):
...
ValueError: Atkin-Lehner only leaves space invariant when character is trivial or quadratic. In general it sends M_k(chi) to M_k(1/chi)
Note that Atkin-Lehner does make sense on `\Gamma_1(13)`,
but doesn't commute with the Hecke operators::
sage: M = ModularSymbols(Gamma1(13),2)
sage: w = M.atkin_lehner_operator(13).matrix()
sage: t = M.T(2).matrix()
sage: t*w == w*t
False
sage: w^2 == 1
True
"""
chi = self.character()
if chi is not None and chi.order() > 2:
raise ValueError, "Atkin-Lehner only leaves space invariant when character is trivial or quadratic. In general it sends M_k(chi) to M_k(1/chi)"
N = self.level()
k = self.weight()
R = self.base_ring()
if N%d != 0:
raise ValueError, "d must divide N"
g, x, y = arith.xgcd(d, -N//d)
g = [d*x, y, N, d]
A = self._action_on_modular_symbols(g)
scale = R(d)**(1 - k//2)
Wmat = scale * A
return Wmat
def boundary_map(self):
r"""
Return the boundary map to the corresponding space of boundary modular
symbols.
EXAMPLES::
sage: ModularSymbols(20,2).boundary_map()
Hecke module morphism boundary map defined by the matrix
[ 1 -1 0 0 0 0]
[ 0 1 -1 0 0 0]
[ 0 1 0 -1 0 0]
[ 0 0 0 -1 1 0]
[ 0 1 0 -1 0 0]
[ 0 0 1 -1 0 0]
[ 0 1 0 0 0 -1]
Domain: Modular Symbols space of dimension 7 for Gamma_0(20) of weight ...
Codomain: Space of Boundary Modular Symbols for Congruence Subgroup Gamma0(20) ...
sage: type(ModularSymbols(20,2).boundary_map())
<class 'sage.modular.hecke.morphism.HeckeModuleMorphism_matrix'>
"""
try:
return self.__boundary_map
except AttributeError:
B = self.boundary_space()
I = [B(b) for b in self.basis()]
W = matrix_space.MatrixSpace(self.base_ring(), len(I), B.rank(), sparse=True)
E = [x.element() for x in I]
zero = self.base_ring()(0)
n = int(B.dimension())
E = sum([ list(x) + [zero]*(n - len(x)) for x in E ], [])
A = W( E )
H = cat.Hom(self, B)
self.__boundary_map = H(A, "boundary map")
return self.__boundary_map
def cusps(self):
r"""
Return the set of cusps for this modular symbols space.
EXAMPLES::
sage: ModularSymbols(20,2).cusps()
[Infinity, 0, -1/4, 1/5, -1/2, 1/10]
"""
try:
return self.__cusps
except AttributeError:
f = self.boundary_map()
B = f.codomain()
C = B._known_cusps()
self.__cusps = C
return C
def boundary_space(self):
r"""
Return the subspace of boundary modular symbols of this modular symbols ambient space.
EXAMPLES::
sage: ModularSymbols(20,2).boundary_space()
Space of Boundary Modular Symbols for Congruence Subgroup Gamma0(20) of weight 2 and over Rational Field
sage: ModularSymbols(20,2).dimension()
7
sage: ModularSymbols(20,2).boundary_space().dimension()
6
"""
raise NotImplementedError
def cuspidal_submodule(self):
"""
The cuspidal submodule of this modular symbols ambient space.
EXAMPLES::
sage: M = ModularSymbols(12,2,0,GF(5)) ; M
Modular Symbols space of dimension 5 for Gamma_0(12) of weight 2 with sign 0 over Finite Field of size 5
sage: M.cuspidal_submodule()
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 5 for Gamma_0(12) of weight 2 with sign 0 over Finite Field of size 5
sage: ModularSymbols(1,24,-1).cuspidal_submodule()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 2 for Gamma_0(1) of weight 24 with sign -1 over Rational Field
The cuspidal submodule of the cuspidal submodule is itself::
sage: M = ModularSymbols(389)
sage: S = M.cuspidal_submodule()
sage: S.cuspidal_submodule() is S
True
"""
try:
return self.__cuspidal_submodule
except AttributeError:
try:
if self.__is_cuspidal:
return self
except AttributeError:
pass
S = self.boundary_map().kernel()
S._set_is_cuspidal(True)
S._is_full_hecke_module = True
if S.dimension() == 1:
S._is_simple = True
if self.base_ring().characteristic() == 0:
d = self._cuspidal_submodule_dimension_formula()
if not d is None:
assert d == S.dimension(), "According to dimension formulas the cuspidal subspace of \"%s\" has dimension %s; however, computing it using modular symbols we obtained %s, so there is a bug (please report!)."%(self, d, S.dimension())
self.__cuspidal_submodule = S
return self.__cuspidal_submodule
def _degeneracy_raising_matrix(self, M, t):
r"""
Return the matrix of the level-raising degeneracy map from self to M,
of index t. This is calculated by composing the level-raising matrix
for `t = 1` with a Hecke operator.
INPUT:
- ``M`` (int) -- a space of modular symbols whose level is an integer
multiple of the the level of self
- ``t`` (int) -- a positive integer dividing the quotient of the two
levels.
OUTPUT:
(matrix) The matrix of the degeneracy map of index `t` from this space
of level `N` to the space `M` (of level a multiple of `N`). Here `t` is
a divisor of the quotient.
EXAMPLES::
sage: A = ModularSymbols(11, 2); B = ModularSymbols(22, 2)
sage: A._degeneracy_raising_matrix(B, 1)
[ 1 0 0 0 0 -1 -1]
[ 0 1 0 -3 1 1 -1]
[ 0 1 1 -1 -1 0 0]
sage: A._degeneracy_raising_matrix(B, 2)
[ 2 0 0 0 1 0 -1]
[ 0 0 -1 3 -1 -1 1]
[ 0 -1 -1 1 0 1 -1]
"""
if t == 1:
return self._degeneracy_raising_matrix_1(M)
else:
d1 = self.degeneracy_map(M, 1).matrix()
T = M.hecke_matrix(t)
return (~self.base_ring()(t)) * d1 * T
def _degeneracy_raising_matrix_1(self, M):
r"""
Return the matrix of the degeneracy map to the given level
(which must be a multiple of the level of self).
.. note::
Not implemented in the base class, only in the derived classes.
EXAMPLES::
sage: M = ModularSymbols(37,4)
sage: M._degeneracy_raising_matrix_1(ModularSymbols(74, 4))
20 x 58 dense matrix over Rational Field
"""
raise NotImplementedError
def _degeneracy_lowering_matrix(self, M, t):
r"""
Return the matrix of the level-lowering degeneracy map from self to M.
INPUT:
- ``M`` -- a modular symbols space whose level divides the level of
self
- ``t`` (int) -- a positive integer dividing the quotient of the
levels.
OUTPUT:
(matrix) The matrix of the degeneracy map from this space to the space
`M` of index `t`, where `t` is a divisor of the quotient of the levels
of self and `M`.
EXAMPLES::
sage: M = ModularSymbols(22,2)
sage: M._degeneracy_lowering_matrix(ModularSymbols(11, 2), 2)
[ 1 0 0]
[ 0 1 -1]
[ 0 0 -1]
[ 0 1 0]
[ 0 0 0]
[-1 0 1]
[-1 0 0]
"""
H = heilbronn.HeilbronnMerel(t)
return self.__heilbronn_operator(M,H,t).matrix()
def rank(self):
"""
Returns the rank of this modular symbols ambient space.
OUTPUT:
(int) The rank of this space of modular symbols.
EXAMPLES::
sage: M = ModularSymbols(389)
sage: M.rank()
65
::
sage: ModularSymbols(11,sign=0).rank()
3
sage: ModularSymbols(100,sign=0).rank()
31
sage: ModularSymbols(22,sign=1).rank()
5
sage: ModularSymbols(1,12).rank()
3
sage: ModularSymbols(3,4).rank()
2
sage: ModularSymbols(8,6,sign=-1).rank()
3
"""
try:
return self.__rank
except AttributeError:
self.__rank = len(self.manin_basis())
return self.__rank
def eisenstein_submodule(self):
"""
Return the Eisenstein submodule of this space of modular symbols.
EXAMPLES::
sage: ModularSymbols(20,2).eisenstein_submodule()
Modular Symbols subspace of dimension 5 of Modular Symbols space of dimension 7 for Gamma_0(20) of weight 2 with sign 0 over Rational Field
"""
try:
return self.__eisenstein_submodule
except AttributeError:
self.__eisenstein_submodule = self.cuspidal_submodule().complement()
return self.__eisenstein_submodule
def element(self, x):
"""
Creates and returns an element of self from a modular symbol, if
possible.
INPUT:
- ``x`` - an object of one of the following types:
ModularSymbol, ManinSymbol.
OUTPUT:
ModularSymbol - a modular symbol with parent self.
EXAMPLES::
sage: M = ModularSymbols(11,4,1)
sage: M.T(3)
Hecke operator T_3 on Modular Symbols space of dimension 4 for Gamma_0(11) of weight 4 with sign 1 over Rational Field
sage: M.T(3)(M.0)
28*[X^2,(0,1)] + 2*[X^2,(1,7)] - [X^2,(1,9)] - [X^2,(1,10)]
sage: M.T(3)(M.0).element()
(28, 2, -1, -1)
"""
if isinstance(x, manin_symbols.ManinSymbol):
if not x.parent().weight() == self.weight():
raise ArithmeticError, "incompatible weights: Manin symbol has weight %s, but modular symbols space has weight %s"%(
x.parent().weight(), self.weight())
t = self.manin_symbols().index(x.tuple())
if isinstance(t, tuple):
i, scalar = t
v = self.manin_gens_to_basis().row(i) * scalar
else:
v = self.manin_gens_to_basis().row(t)
return element.ModularSymbolsElement(self, v)
elif isinstance(x, element.ModularSymbolsElement):
M = x.parent()
if M.ambient_hecke_module() != self:
raise TypeError, "Modular symbol (%s) does not lie in this space."%x
return self(x.element())
else:
raise ValueError, "Cannot create element of %s from %s."%(x,self)
def dual_star_involution_matrix(self):
"""
Return the matrix of the dual star involution, which is induced by
complex conjugation on the linear dual of modular symbols.
EXAMPLES::
sage: ModularSymbols(20,2).dual_star_involution_matrix()
[1 0 0 0 0 0 0]
[0 1 0 0 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 1 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 0 0 1 0]
[0 0 0 0 0 0 1]
"""
try:
return self.__dual_star_involution_matrix
except AttributeError:
pass
self.__dual_star_involution_matrix = self.star_involution().matrix().transpose()
return self.__dual_star_involution_matrix
def factorization(self):
r"""
Returns a list of pairs `(S,e)` where `S` is spaces
of modular symbols and self is isomorphic to the direct sum of the
`S^e` as a module over the *anemic* Hecke algebra adjoin
the star involution. The cuspidal `S` are all simple, but
the Eisenstein factors need not be simple.
EXAMPLES::
sage: ModularSymbols(Gamma0(22), 2).factorization()
(Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field)^2 *
(Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field)^2 *
(Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 7 for Gamma_0(22) of weight 2 with sign 0 over Rational Field)
::
sage: ModularSymbols(1,6,0,GF(2)).factorization()
(Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(1) of weight 6 with sign 0 over Finite Field of size 2) *
(Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(1) of weight 6 with sign 0 over Finite Field of size 2)
::
sage: ModularSymbols(18,2).factorization()
(Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 7 for Gamma_0(18) of weight 2 with sign 0 over Rational Field) *
(Modular Symbols subspace of dimension 5 of Modular Symbols space of dimension 7 for Gamma_0(18) of weight 2 with sign 0 over Rational Field)
::
sage: M = ModularSymbols(DirichletGroup(38,CyclotomicField(3)).0^2, 2, +1); M
Modular Symbols space of dimension 7 and level 38, weight 2, character [zeta3], sign 1, over Cyclotomic Field of order 3 and degree 2
sage: M.factorization() # long time (about 8 seconds)
(Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 7 and level 38, weight 2, character [zeta3], sign 1, over Cyclotomic Field of order 3 and degree 2) *
(Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 7 and level 38, weight 2, character [zeta3], sign 1, over Cyclotomic Field of order 3 and degree 2) *
(Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 7 and level 38, weight 2, character [zeta3], sign 1, over Cyclotomic Field of order 3 and degree 2) *
(Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 7 and level 38, weight 2, character [zeta3], sign 1, over Cyclotomic Field of order 3 and degree 2)
"""
try:
return self._factorization
except AttributeError:
pass
try:
if self._is_simple:
return [self]
except AttributeError:
pass
D = []
k = self.weight()
if self.base_ring().characteristic() == 2:
skip_minus = True
else:
skip_minus = False
chi = self.character()
cond = 1 if chi is None else chi.conductor()
for d in reversed(arith.divisors(self.level())):
if d%cond != 0: continue
n = arith.number_of_divisors(self.level() // d)
M = self.modular_symbols_of_level(d)
N = M.new_submodule().cuspidal_submodule().decomposition()
for A in N:
if self.sign() == 0:
V = A.plus_submodule()
V._is_simple = True
D.append((V,n))
if skip_minus:
continue
V = A.minus_submodule()
V._is_simple = True
D.append((V,n))
else:
A._is_simple = True
D.append((A,n))
for E in self.eisenstein_submodule().decomposition(anemic=True):
D.append((E,1))
r = self.dimension()
s = sum([A.rank()*mult for A, mult in D])
D = sage.structure.all.Factorization(D, cr=True, sort=False)
D.sort(_cmp = cmp)
assert r == s, "bug in factorization -- self has dimension %s, but sum of dimensions of factors is %s\n%s"%(r, s, D)
self._factorization = D
return self._factorization
factor = factorization
def is_cuspidal(self):
r"""
Returns True if this space is cuspidal, else False.
EXAMPLES::
sage: M = ModularSymbols(20,2)
sage: M.is_cuspidal()
False
sage: S = M.cuspidal_subspace()
sage: S.is_cuspidal()
True
sage: S = M.eisenstein_subspace()
sage: S.is_cuspidal()
False
"""
try:
return self.__is_cuspidal
except AttributeError:
S = self.ambient_hecke_module().cuspidal_submodule()
self.__is_cuspidal = (S.dimension() == self.dimension())
return self.__is_cuspidal
def is_eisenstein(self):
r"""
Returns True if this space is Eisenstein, else False.
EXAMPLES::
sage: M = ModularSymbols(20,2)
sage: M.is_eisenstein()
False
sage: S = M.eisenstein_submodule()
sage: S.is_eisenstein()
True
sage: S = M.cuspidal_subspace()
sage: S.is_eisenstein()
False
"""
try:
return self.__is_eisenstein
except AttributeError:
S = self.ambient_hecke_module().eisenstein_submodule()
self.__is_eisenstein = self.dimension()==S.dimension()
return self.__is_eisenstein
def manin_symbols_basis(self):
"""
A list of Manin symbols that form a basis for the ambient space
self. INPUT:
- ``ModularSymbols self`` - an ambient space of
modular symbols
OUTPUT:
- ``list`` - a list of 2-tuples (if the weight is 2)
or 3-tuples, which represent the Manin symbols basis for self.
EXAMPLES::
sage: m = ModularSymbols(23)
sage: m.manin_symbols_basis()
[(1,0), (1,17), (1,19), (1,20), (1,21)]
sage: m = ModularSymbols(6, weight=4, sign=-1)
sage: m.manin_symbols_basis()
[[X^2,(2,1)]]
"""
s = self.manin_symbols()
return [s.manin_symbol(i) for i in self.manin_basis()]
def modular_symbols_of_sign(self, sign):
r"""
Returns a space of modular symbols with the same defining
properties (weight, level, etc.) as this space except with given
sign.
INPUT:
- ``sign`` (int) -- A sign (`+1`, `-1` or `0`).
OUTPUT:
(ModularSymbolsAmbient) A space of modular symbols with the
same defining properties (weight, level, etc.) as this space
except with given sign.
EXAMPLES::
sage: M = ModularSymbols(Gamma0(11),2,sign=0)
sage: M
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: M.modular_symbols_of_sign(-1)
Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field
sage: M = ModularSymbols(Gamma1(11),2,sign=0)
sage: M.modular_symbols_of_sign(-1)
Modular Symbols space of dimension 1 for Gamma_1(11) of weight 2 with sign -1 and over Rational Field
"""
if sign == self.sign():
return self
return modsym.ModularSymbols(self.group(), self.weight(), sign=sign, base_ring=self.base_ring())
def modular_symbols_of_weight(self, k):
r"""
Returns a space of modular symbols with the same defining
properties (weight, sign, etc.) as this space except with weight
`k`.
INPUT:
- ``k`` (int) -- A positive integer.
OUTPUT:
(ModularSymbolsAmbient) A space of modular symbols with the
same defining properties (level, sign) as this space
except with given weight.
EXAMPLES::
sage: M = ModularSymbols(Gamma1(6),2,sign=0)
sage: M.modular_symbols_of_weight(3)
Modular Symbols space of dimension 4 for Gamma_1(6) of weight 3 with sign 0 and over Rational Field
"""
if k == self.weight():
return self
return modsym.ModularSymbols(self.group(), weight=k, sign=self.sign(), base_ring=self.base_ring())
def _compute_sign_submodule(self, sign, compute_dual=True):
r"""
Return the subspace of self that is fixed under the star
involution.
INPUT:
- ``sign`` - int (either -1 or +1)
- ``compute_dual`` - bool (default: True) also
compute dual subspace. This are useful for many algorithms.
OUTPUT:
A subspace of modular symbols
EXAMPLES::
sage: ModularSymbols(1,12,0,GF(5)).minus_submodule() ## indirect doctest
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Finite Field of size 5
"""
S = self.star_involution().matrix() - self.base_ring()(sign)
V = S.kernel()
if compute_dual:
Vdual = S.transpose().kernel()
M = self.submodule(V, Vdual, check=False)
else:
M = self.submodule(V, check=False)
M._set_sign(sign)
return M
def star_involution(self):
r"""
Return the star involution on this modular symbols space.
OUTPUT:
(matrix) The matrix of the star involution on this space,
which is induced by complex conjugation on modular symbols,
with respect to the standard basis.
EXAMPLES::
sage: ModularSymbols(20,2).star_involution()
Hecke module morphism Star involution on Modular Symbols space of dimension 7 for Gamma_0(20) of weight 2 with sign 0 over Rational Field defined by the matrix
[1 0 0 0 0 0 0]
[0 1 0 0 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 1 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 0 0 1 0]
[0 0 0 0 0 0 1]
Domain: Modular Symbols space of dimension 7 for Gamma_0(20) of weight ...
Codomain: Modular Symbols space of dimension 7 for Gamma_0(20) of weight ...
"""
try:
return self.__star_involution
except AttributeError:
pass
S = self.__heilbronn_operator(self, [[-1,0, 0,1]], 1)
S.name("Star involution on %s"%self)
self.__star_involution = S
return self.__star_involution
def _compute_diamond_matrix(self, d):
r"""
Return the diamond bracket d operator on this modular symbols space.
INPUT:
- `d` -- integer
OUTPUT:
- ``matrix`` - the matrix of the diamond bracket operator
on this space.
EXAMPLES::
sage: e = kronecker_character(7)
sage: M = ModularSymbols(e,2,sign=1)
sage: D = M.diamond_bracket_operator(5); D
Diamond bracket operator <5> on Modular Symbols space ...
sage: D.matrix() # indirect doctest
[-1 0 0 0]
[ 0 -1 0 0]
[ 0 0 -1 0]
[ 0 0 0 -1]
sage: [M.diamond_bracket_operator(d).matrix()[0,0] for d in [0..6]]
[0, 1, 0, 1, 0, -1, 0]
sage: [e(d) for d in [0..6]]
[0, 1, 0, 1, 0, -1, 0]
We test that the sign issue at #8620 is fixed::
sage: M = Newforms(Gamma1(13),names = 'a')[0].modular_symbols(sign=0)
sage: M.diamond_bracket_operator(4).matrix()
[ 0 0 1 -1]
[-1 -1 0 1]
[-1 -1 0 0]
[ 0 -1 1 -1]
We check that the result is correctly normalised for weight > 2::
sage: ModularSymbols(Gamma1(13), 5).diamond_bracket_operator(6).charpoly().factor()
(x^2 + 1)^8 * (x^4 - x^2 + 1)^10
"""
return self.__heilbronn_operator(self, [[d,0, 0,d]], 1).matrix() * d**(2 - self.weight())
def submodule(self, M, dual_free_module=None, check=True):
r"""
Return the submodule with given generators or free module `M`.
INPUT:
- ``M`` - either a submodule of this ambient free module, or
generators for a submodule;
- ``dual_free_module`` (bool, default None) -- this may be
useful to speed up certain calculations; it is the
corresponding submodule of the ambient dual module;
- ``check`` (bool, default True) -- if True, check that `M` is
a submodule, i.e. is invariant under all Hecke operators.
OUTPUT:
A subspace of this modular symbol space.
EXAMPLES::
sage: M = ModularSymbols(11)
sage: M.submodule([M.0])
Traceback (most recent call last):
...
ValueError: The submodule must be invariant under all Hecke operators.
sage: M.eisenstein_submodule().basis()
((1,0) - 1/5*(1,9),)
sage: M.basis()
((1,0), (1,8), (1,9))
sage: M.submodule([M.0 - 1/5*M.2])
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
.. note::
It would make more sense to only check that `M` is invariant
under the Hecke operators with index coprime to the level.
Unfortunately, I do not know a reasonable algorithm for
determining whether a module is invariant under just the
anemic Hecke algebra, since I do not know an analogue of
the Sturm bound for the anemic Hecke algebra. - William
Stein, 2007-07-27
"""
if check:
if not free_module.is_FreeModule(M):
V = self.free_module()
if not isinstance(M, (list,tuple)):
M = M.gens()
M = V.span([V(x.element()) for x in M])
return subspace.ModularSymbolsSubspace(self, M, dual_free_module=dual_free_module, check=check)
def twisted_winding_element(self, i, eps):
r"""
Return the twisted winding element of given degree and character.
INPUT:
- ``i`` (int) -- an integer, `0\le i\le k-2` where `k` is the weight.
- ``eps`` (character) -- a Dirichlet character
OUTPUT:
(modular symbol) The so-called 'twisted winding element':
.. math::
\sum_{a \in (\ZZ/m\ZZ)^\times} \varepsilon(a) * [ i, 0, a/m ].
.. note::
This will only work if the base ring of the modular symbol
space contains the character values.
EXAMPLES::
sage: eps = DirichletGroup(5)[2]
sage: K = eps.base_ring()
sage: M = ModularSymbols(37,2,0,K)
sage: M.twisted_winding_element(0,eps)
2*(1,23) + (-2)*(1,32) + 2*(1,34)
"""
if not dirichlet.is_DirichletCharacter(eps):
raise TypeError, "eps must be a Dirichlet character."
if (i < 0) or (i > self.weight()-2):
raise ValueError, "i must be between 0 and k-2."
m = eps.modulus()
s = self(0)
for a in ([ x for x in range(1,m) if rings.gcd(x,m) == 1 ]):
s += eps(a) * self.modular_symbol([i, Cusp(0), Cusp(a/m)])
return s
def integral_structure(self, algorithm='default'):
r"""
Return the `\ZZ`-structure of this modular symbols
space, generated by all integral modular symbols.
INPUT:
- ``algorithm`` - string (default: 'default' - choose
heuristically)
- ``'pari'`` - use pari for the HNF computation
- ``'padic'`` - use p-adic algorithm (only good for
dense case)
ALGORITHM: It suffices to consider lattice generated by the free
generating symbols `X^iY^{k-2-i}.(u,v)` after quotienting
out by the `S` (and `I`) relations, since the
quotient by these relations is the same over any ring.
EXAMPLES: In weight 2 the rational basis is often integral.
::
sage: M = ModularSymbols(11,2)
sage: M.integral_structure()
Free module of degree 3 and rank 3 over Integer Ring
Echelon basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]
This is rarely the case in higher weight::
sage: M = ModularSymbols(6,4)
sage: M.integral_structure()
Free module of degree 6 and rank 6 over Integer Ring
Echelon basis matrix:
[ 1 0 0 0 0 0]
[ 0 1 0 0 0 0]
[ 0 0 1/2 1/2 1/2 1/2]
[ 0 0 0 1 0 0]
[ 0 0 0 0 1 0]
[ 0 0 0 0 0 1]
Here is an example involving `\Gamma_1(N)`.
::
sage: M = ModularSymbols(Gamma1(5),6)
sage: M.integral_structure()
Free module of degree 10 and rank 10 over Integer Ring
Echelon basis matrix:
[ 1 0 0 0 0 0 0 0 0 0]
[ 0 1 0 0 0 0 0 0 0 0]
[ 0 0 1/102 0 5/204 1/136 23/24 3/17 43/136 69/136]
[ 0 0 0 1/48 0 1/48 23/24 1/6 1/8 17/24]
[ 0 0 0 0 1/24 0 23/24 1/3 1/6 1/2]
[ 0 0 0 0 0 1/24 23/24 1/3 11/24 5/24]
[ 0 0 0 0 0 0 1 0 0 0]
[ 0 0 0 0 0 0 0 1/2 0 1/2]
[ 0 0 0 0 0 0 0 0 1/2 1/2]
[ 0 0 0 0 0 0 0 0 0 1]
"""
if not self.base_ring() == rational_field.RationalField():
raise NotImplementedError
try:
return self.__integral_structure
except AttributeError:
pass
G = set([i for i, _ in self._mod2term])
G = list(G)
G.sort()
B = self._manin_gens_to_basis.matrix_from_rows(list(G)).dense_matrix()
B, d = B._clear_denom()
if algorithm == 'default':
if self.weight() == 2:
algorithm = 'pari'
else:
algorithm = 'padic'
if algorithm == 'pari':
B = B.echelon_form(algorithm='pari', include_zero_rows=False)
elif algorithm == 'padic':
B = B.echelon_form(algorithm='padic', include_zero_rows=False)
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
W = B.row_module()
if d != 1:
W = W.scale(1/d)
self.__integral_structure = W
assert W.rank() == self.rank(), "there is a bug in computing integral structure on modular symbols"
return self.__integral_structure
def compact_newform_eigenvalues(self, v, names='alpha'):
r"""
Return compact systems of eigenvalues for each Galois conjugacy
class of cuspidal newforms in this ambient space.
INPUT:
- ``v`` - list of positive integers
OUTPUT:
- ``list`` - of pairs (E, x), where E\*x is a vector
with entries the eigenvalues `a_n` for
`n \in v`.
EXAMPLES::
sage: M = ModularSymbols(43,2,1)
sage: X = M.compact_newform_eigenvalues(prime_range(10))
sage: X[0][0] * X[0][1]
(-2, -2, -4, 0)
sage: X[1][0] * X[1][1]
(alpha1, -alpha1, -alpha1 + 2, alpha1 - 2)
::
sage: M = ModularSymbols(DirichletGroup(24,QQ).1,2,sign=1)
sage: M.compact_newform_eigenvalues(prime_range(10),'a')
[([-1/2 -1/2]
[ 1/2 -1/2]
[ -1 1]
[ -2 0], (1, -2*a0 - 1))]
sage: a = M.compact_newform_eigenvalues([1..10],'a')[0]
sage: a[0]*a[1]
(1, a0, a0 + 1, -2*a0 - 2, -2*a0 - 2, -a0 - 2, -2, 2*a0 + 4, -1, 2*a0 + 4)
sage: M = ModularSymbols(DirichletGroup(13).0^2,2,sign=1)
sage: M.compact_newform_eigenvalues(prime_range(10),'a')
[([ -zeta6 - 1]
[ 2*zeta6 - 2]
[-2*zeta6 + 1]
[ 0], (1))]
sage: a = M.compact_newform_eigenvalues([1..10],'a')[0]
sage: a[0]*a[1]
(1, -zeta6 - 1, 2*zeta6 - 2, zeta6, -2*zeta6 + 1, -2*zeta6 + 4, 0, 2*zeta6 - 1, -zeta6, 3*zeta6 - 3)
"""
if self.sign() == 0:
raise ValueError, "sign must be nonzero"
v = list(v)
D = self.cuspidal_submodule().new_subspace().decomposition()
for A in D:
A._is_simple = True
B = [A.dual_free_module().basis_matrix().transpose() for A in D]
names = ['%s%s'%(names,i) for i in range(len(B))]
nz = None
for i in range(self.dimension()):
bad = False
for C in B:
if C.row(i) == 0:
bad = True
continue
if bad: continue
nz = i
break
if nz is not None:
R = self.hecke_images(nz, v)
return [(R*m, D[i].dual_eigenvector(names=names[i], lift=False, nz=nz)) for i, m in enumerate(B)]
else:
ans = []
cache = {}
for i in range(len(D)):
nz = D[i]._eigen_nonzero()
if cache.has_key(nz):
R = cache[nz]
else:
R = self.hecke_images(nz, v)
cache[nz] = R
ans.append((R*B[i], D[i].dual_eigenvector(names=names[i], lift=False, nz=nz)))
return ans
class ModularSymbolsAmbient_wtk_g0(ModularSymbolsAmbient):
r"""
Modular symbols for `\Gamma_0(N)` of integer weight
`k > 2` over the field `F`.
For weight `2`, it is faster to use ``ModularSymbols_wt2_g0``.
INPUT:
- ``N`` - int, the level
- ``k`` - integer weight = 2.
- ``sign`` - int, either -1, 0, or 1
- ``F`` - field
EXAMPLES::
sage: ModularSymbols(1,12)
Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
sage: ModularSymbols(1,12, sign=1).dimension()
2
sage: ModularSymbols(15,4, sign=-1).dimension()
4
sage: ModularSymbols(6,6).dimension()
10
sage: ModularSymbols(36,4).dimension()
36
"""
def __init__(self, N, k, sign, F, custom_init=None):
r"""
Initialize a space of modular symbols of weight `k` for
`\Gamma_0(N)`, over `\QQ`.
For weight `2`, it is faster to use
``ModularSymbols_wt2_g0``.
INPUT:
- ``N`` - int, the level
- ``k`` - integer weight = 2.
- ``sign`` - int, either -1, 0, or 1
- ``F`` - field
EXAMPLES::
sage: ModularSymbols(1,12)
Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
sage: ModularSymbols(1,12, sign=1).dimension()
2
sage: ModularSymbols(15,4, sign=-1).dimension()
4
sage: ModularSymbols(6,6).dimension()
10
sage: ModularSymbols(36,4).dimension()
36
"""
N = int(N)
k = int(k)
sign = int(sign)
if not sign in [-1,0,1]:
raise TypeError, "sign must be an int in [-1,0,1]"
ModularSymbolsAmbient.__init__(self, weight=k, group=arithgroup.Gamma0(N),
sign=sign, base_ring=F, custom_init=custom_init)
def _dimension_formula(self):
r"""
Return the dimension of this space using the formula.
EXAMPLES::
sage: M = ModularSymbols(37,6)
sage: M.dimension()
32
sage: M._dimension_formula()
32
"""
if self.base_ring().characteristic() == 0:
N, k, sign = self.level(), self.weight(), self.sign()
if sign != 0: return None
if k%2 == 1:
return 0
elif k > 2:
return 2*self.group().dimension_cusp_forms(k) + self.group().ncusps()
else:
return 2*self.group().dimension_cusp_forms(k) + self.group().ncusps() - 1
else:
raise NotImplementedError
def _repr_(self):
r"""
Return the string representation of this Modular Symbols space.
EXAMPLES::
sage: M = ModularSymbols(37,6)
sage: M # indirect doctest
Modular Symbols space of dimension 32 for Gamma_0(37) of weight 6 with sign 0 over Rational Field
"""
return ("Modular Symbols space of dimension %s for Gamma_0(%s) of weight %s with sign %s " + \
"over %s")%(self.dimension(), self.level(),self.weight(), self.sign(),
self.base_ring())
def _cuspidal_submodule_dimension_formula(self):
r"""
Return the dimension of the cuspidal subspace, using the formula.
EXAMPLES::
sage: M = ModularSymbols(37,4)
sage: M.cuspidal_subspace().dimension()
18
sage: M._cuspidal_submodule_dimension_formula()
18
"""
if self.base_ring().characteristic() == 0:
N, k, sign = self.level(), self.weight(), self.sign()
if sign == 0:
m = 2
else:
m = 1
return m * self.group().dimension_cusp_forms(k)
else:
raise NotImplementedError
def _degeneracy_raising_matrix_1(self, M):
r"""
Return the matrix of the degeneracy map (with t = 1) to level
`N`, where `N` is a multiple of the level.
INPUT:
- ``M`` -- A space of Gamma0 modular symbols of the same weight as
self, with level an integer multiple of the level of self.
OUTPUT:
(matrix) The matrix of the degeneracy raising map to `M`.
EXAMPLES::
sage: M = ModularSymbols(37,4)
sage: M._degeneracy_raising_matrix_1(ModularSymbols(74, 4))
20 x 58 dense matrix over Rational Field
sage: M.dimension()
20
sage: ModularSymbols(74,4).dimension()
58
"""
level = int(M.level())
N = self.level()
H = arithgroup.degeneracy_coset_representatives_gamma0(level, N, 1)
MS = matrix_space.MatrixSpace(self.base_ring(), self.dimension(), M.dimension())
if self.dimension() == 0 or M.dimension() == 0:
return MS(0)
rows = []
B = self.manin_basis()
syms = self.manin_symbols()
k = self.weight()
G = matrix_space.MatrixSpace(integer_ring.IntegerRing(),2)
H = [G(h) for h in H]
for n in B:
z = M(0)
s = syms.manin_symbol(n)
g = G(list(s.lift_to_sl2z(N)))
i = s.i
for h in H:
hg = h*g
z += M((i, hg[1,0], hg[1,1]))
rows.append(z.element())
A = MS(rows)
return A
def _cuspidal_new_submodule_dimension_formula(self):
r"""
Return the dimension of the new cuspidal subspace, via the formula.
EXAMPLES:
sage: M = ModularSymbols(100,2)
sage: M._cuspidal_new_submodule_dimension_formula()
2
sage: M.cuspidal_subspace().new_subspace().dimension()
2
"""
if self.base_ring().characteristic() == 0:
N, k, sign = self.level(), self.weight(), self.sign()
if sign == 0:
m = 2
else:
m = 1
return m * self.group().dimension_new_cusp_forms(k)
else:
raise NotImplementedError
def boundary_space(self):
r"""
Return the space of boundary modular symbols for this space.
EXAMPLES::
sage: M = ModularSymbols(100,2)
sage: M.boundary_space()
Space of Boundary Modular Symbols for Congruence Subgroup Gamma0(100) of weight 2 and over Rational Field
"""
try:
return self.__boundary_space
except AttributeError:
pass
self.__boundary_space = boundary.BoundarySpace_wtk_g0(
self.level(), self.weight(), self.sign(), self.base_ring())
return self.__boundary_space
def manin_symbols(self):
r"""
Return the Manin symbol list of this modular symbol space.
EXAMPLES::
sage: M = ModularSymbols(100,4)
sage: M.manin_symbols()
Manin Symbol List of weight 4 for Gamma0(100)
sage: len(M.manin_symbols())
540
"""
try:
return self.__manin_symbols
except AttributeError:
self.__manin_symbols = manin_symbols.ManinSymbolList_gamma0(
level=self.level(), weight=self.weight())
return self.__manin_symbols
def modular_symbols_of_level(self, N):
r"""
Returns a space of modular symbols with the same parameters as
this space except with level `N`.
INPUT:
- ``N`` (int) -- a positive integer.
OUTPUT:
(Modular Symbol space) A space of modular symbols with the
same defining properties (weight, sign, etc.) as this space
except with level `N`.
For example, if self is the space of modular symbols of weight `2`
for `\Gamma_0(22)`, and level is `11`, then this function returns
the modular symbol space of weight `2` for `\Gamma_0(11)`.
EXAMPLES::
sage: M = ModularSymbols(11)
sage: M.modular_symbols_of_level(22)
Modular Symbols space of dimension 7 for Gamma_0(22) of weight 2 with sign 0 over Rational Field
sage: M = ModularSymbols(Gamma1(6))
sage: M.modular_symbols_of_level(12)
Modular Symbols space of dimension 9 for Gamma_1(12) of weight 2 with sign 0 and over Rational Field
"""
return modsym.ModularSymbols(arithgroup.Gamma0(rings.Integer(N)),
self.weight(), sign=self.sign(),
base_ring=self.base_ring())
def _hecke_images(self, i, v):
"""
Return matrix whose rows are the images of the `i`-th
standard basis vector under the Hecke operators `T_p` for
all integers in `v`.
INPUT:
- ``i`` - nonnegative integer
- ``v`` - a list of positive integer
OUTPUT:
- ``matrix`` - whose rows are the Hecke images
EXAMPLES::
sage: M = ModularSymbols(11,4,1)
sage: M._hecke_images(0,[1,2,3,4])
[ 1 0 0 0]
[ 9 0 1 -1]
[28 2 -1 -1]
[73 2 5 -7]
sage: M.T(1)(M.0).element()
(1, 0, 0, 0)
sage: M.T(2)(M.0).element()
(9, 0, 1, -1)
sage: M.T(3)(M.0).element()
(28, 2, -1, -1)
sage: M.T(4)(M.0).element()
(73, 2, 5, -7)
sage: M = ModularSymbols(12,4)
sage: M._hecke_images(0,[1,2,3,4])
[ 1 0 0 0 0 0 0 0 0 0 0 0]
[ 8 1 -1 -2 2 2 -3 1 -2 3 -1 0]
[ 27 4 -4 -8 8 10 -14 4 -9 14 -5 0]
[ 64 10 -10 -20 20 26 -36 10 -24 38 -14 0]
sage: M.T(1)(M.0).element()
(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
sage: M.T(2)(M.0).element()
(8, 1, -1, -2, 2, 2, -3, 1, -2, 3, -1, 0)
sage: M.T(3)(M.0).element()
(27, 4, -4, -8, 8, 10, -14, 4, -9, 14, -5, 0)
sage: M.T(4)(M.0).element()
(64, 10, -10, -20, 20, 26, -36, 10, -24, 38, -14, 0)
"""
c = self.manin_generators()[self.manin_basis()[i]]
N = self.level()
return heilbronn.hecke_images_gamma0_weight_k(c.u,c.v, c.i, N, self.weight(),
v, self.manin_gens_to_basis())
class ModularSymbolsAmbient_wt2_g0(ModularSymbolsAmbient_wtk_g0):
r"""
Modular symbols for `\Gamma_0(N)` of integer weight `2` over the field
`F`.
INPUT:
- ``N`` - int, the level
- ``sign`` - int, either -1, 0, or 1
OUTPUT:
The space of modular symbols of weight `2`, trivial character, level
`N` and given sign.
EXAMPLES::
sage: ModularSymbols(Gamma0(12),2)
Modular Symbols space of dimension 5 for Gamma_0(12) of weight 2 with sign 0 over Rational Field
"""
def __init__(self, N, sign, F, custom_init=None):
"""
Initialize a space of modular symbols. INPUT:
INPUT:
- ``N`` - int, the level
- ``sign`` - int, either -1, 0, or 1
OUTPUT:
The space of modular symbols of weight 2, trivial character,
level N and given sign.
EXAMPLES::
sage: M = ModularSymbols(Gamma0(12),2)
"""
ModularSymbolsAmbient_wtk_g0.__init__(self,
N=N, k=2, sign=sign, F=F,
custom_init=custom_init)
def _dimension_formula(self):
r"""
Return the dimension of this space using the formula.
EXAMPLES::
sage: M = ModularSymbols(37,6)
sage: M.dimension()
32
sage: M._dimension_formula()
32
"""
if self.base_ring().characteristic() == 0:
N, sign = self.level(), self.sign()
if sign != 0: return None
return 2*self.group().dimension_cusp_forms(2) + self.group().ncusps() - 1
else:
raise NotImplementedError
def _cuspidal_submodule_dimension_formula(self):
r"""
Return the dimension of the cuspidal subspace, using the formula.
EXAMPLES::
sage: M = ModularSymbols(37,4)
sage: M.cuspidal_subspace().dimension()
18
sage: M._cuspidal_submodule_dimension_formula()
18
"""
if self.base_ring().characteristic() == 0:
if self.sign() == 0:
m = 2
else:
m = 1
return m * self.group().dimension_cusp_forms(2)
else:
raise NotImplementedError
def _cuspidal_new_submodule_dimension_formula(self):
r"""
Return the dimension of the new cuspidal subspace, via the formula.
EXAMPLES:
sage: M = ModularSymbols(100,2)
sage: M._cuspidal_new_submodule_dimension_formula()
2
sage: M.cuspidal_subspace().new_subspace().dimension()
2
"""
if self.base_ring().characteristic() == 0:
if self.sign() == 0:
m = 2
else:
m = 1
return m * self.group().dimension_new_cusp_forms(2)
else:
raise NotImplementedError
def _compute_hecke_matrix_prime(self, p, rows=None):
r"""
Compute and return the matrix of the `p`-th Hecke operator.
EXAMPLES::
sage: m = ModularSymbols(37,2)
sage: m._compute_hecke_matrix_prime(2).charpoly('x')
x^5 + x^4 - 8*x^3 - 12*x^2
"""
if isinstance(rows, list):
rows = tuple(rows)
try:
return self._hecke_matrices[(p,rows)]
except AttributeError:
self._hecke_matrices = {}
except KeyError:
pass
tm = misc.verbose("Computing Hecke operator T_%s"%p)
H = heilbronn.HeilbronnCremona(p)
B = self.manin_basis()
if not rows is None:
B = [B[i] for i in rows]
cols = []
N = self.level()
P1 = self.p1list()
mod2term = self._mod2term
R = self.manin_gens_to_basis()
W = R.new_matrix(nrows=len(B), ncols = R.nrows())
j = 0
tm = misc.verbose("Matrix non-reduced", tm)
for i in B:
c,d = P1[i]
v = H.apply(c,d, N)
for z, m in v:
k = P1.index_of_normalized_pair(z[0],z[1])
if k != -1:
f, s = mod2term[k]
if s != 0:
W[j,f] = W[j,f] + s*m
j += 1
tm = misc.verbose("done making non-reduced matrix",tm)
misc.verbose("start matrix-matrix (%s x %s) times (%s x %s) multiply to get Tp"%(W.nrows(), W.ncols(),
R.nrows(), R.ncols()))
if hasattr(W, '_matrix_times_matrix_dense'):
Tp = W._matrix_times_matrix_dense(R)
misc.verbose("done matrix multiply and computing Hecke operator",tm)
else:
Tp = W * R
tm = misc.verbose("done multiplying",tm)
Tp = Tp.dense_matrix()
misc.verbose("done making hecke operator dense",tm)
if rows is None:
self._hecke_matrices[(p,rows)] = Tp
return Tp
def boundary_space(self):
r"""
Return the space of boundary modular symbols for this space.
EXAMPLES::
sage: M = ModularSymbols(100,2)
sage: M.boundary_space()
Space of Boundary Modular Symbols for Congruence Subgroup Gamma0(100) of weight 2 and over Rational Field
"""
try:
return self.__boundary_space
except AttributeError:
pass
self.__boundary_space = boundary.BoundarySpace_wtk_g0(
self.level(), self.weight(), self.sign(), self.base_ring())
return self.__boundary_space
def _hecke_image_of_ith_basis_vector(self, n, i):
"""
Return `T_n(e_i)`, where `e_i` is the
`i`th basis vector of this ambient space.
INPUT:
- ``n`` - an integer which should be prime.
OUTPUT:
- ``modular symbol`` - element of this ambient space
EXAMPLES::
sage: M = ModularSymbols(43,2,1)
sage: M._hecke_image_of_ith_basis_vector(2, 0)
3*(1,0) - 2*(1,33)
sage: M.hecke_operator(2)(M.0)
3*(1,0) - 2*(1,33)
sage: M._hecke_image_of_ith_basis_vector(6, 1)
-2*(1,33)
sage: M.hecke_operator(6)(M.1)
-2*(1,33)
"""
c = self.manin_generators()[self.manin_basis()[i]]
N = self.level()
I = heilbronn.hecke_images_gamma0_weight2(c.u,c.v,N,[n], self.manin_gens_to_basis())
return self(I[0])
def _hecke_images(self, i, v):
"""
Return images of the `i`-th standard basis vector under the
Hecke operators `T_p` for all integers in `v`.
INPUT:
- ``i`` - nonnegative integer
- ``v`` - a list of positive integer
OUTPUT:
- ``matrix`` - whose rows are the Hecke images
EXAMPLES::
sage: M = ModularSymbols(46,2,-1)
sage: M._hecke_images(1,[3,4,5,6])
[ 0 1 -2 2 0]
[ 2 -3 0 0 1]
[ 0 -2 2 -2 0]
[-5 3 -1 1 0]
sage: v = M.basis()[1]
sage: M.T(3)(v).element()
(0, 1, -2, 2, 0)
sage: M.T(4)(v).element()
(2, -3, 0, 0, 1)
sage: M.T(5)(v).element()
(0, -2, 2, -2, 0)
sage: M.T(6)(v).element()
(-5, 3, -1, 1, 0)
"""
c = self.manin_generators()[self.manin_basis()[i]]
N = self.level()
return heilbronn.hecke_images_gamma0_weight2(c.u,c.v,N, v, self.manin_gens_to_basis())
class ModularSymbolsAmbient_wtk_g1(ModularSymbolsAmbient):
r"""
INPUT:
- ``level`` - int, the level
- ``weight`` - int, the weight = 2
- ``sign`` - int, either -1, 0, or 1
- ``F`` - field
EXAMPLES::
sage: ModularSymbols(Gamma1(17),2)
Modular Symbols space of dimension 25 for Gamma_1(17) of weight 2 with sign 0 and over Rational Field
sage: [ModularSymbols(Gamma1(7),k).dimension() for k in [2,3,4,5]]
[5, 8, 12, 16]
::
sage: ModularSymbols(Gamma1(7),3)
Modular Symbols space of dimension 8 for Gamma_1(7) of weight 3 with sign 0 and over Rational Field
"""
def __init__(self, level, weight, sign, F, custom_init=None):
r"""
Initialize a space of modular symbols for Gamma1(N).
INPUT:
- ``level`` - int, the level
- ``weight`` - int, the weight = 2
- ``sign`` - int, either -1, 0, or 1
- ``F`` - field
EXAMPLES::
sage: ModularSymbols(Gamma1(17),2)
Modular Symbols space of dimension 25 for Gamma_1(17) of weight 2 with sign 0 and over Rational Field
sage: [ModularSymbols(Gamma1(7),k).dimension() for k in [2,3,4,5]]
[5, 8, 12, 16]
::
sage: M = ModularSymbols(Gamma1(7),3)
"""
ModularSymbolsAmbient.__init__(self,
weight=weight,
group=arithgroup.Gamma1(level),
sign=sign,
base_ring=F,
custom_init=custom_init)
def _dimension_formula(self):
r"""
Return the dimension of this space using the formula.
EXAMPLES::
sage: M = ModularSymbols(Gamma1(7),6)
sage: M.dimension()
20
sage: M._dimension_formula()
20
"""
if self.base_ring().characteristic() != 0:
raise NotImplementedError
level, weight, sign = self.level(), self.weight(), self.sign()
if sign != 0: return None
d = 2*self.group().dimension_cusp_forms(weight) + self.group().ncusps()
if level == 1 and weight%2 == 1:
return 0
if weight == 2:
return d - 1
if weight % 2 == 0:
return d
return None
def _repr_(self):
r"""
Return a string representation of this space.
EXAMPLES::
sage: M = ModularSymbols(Gamma1(7),3)
sage: M # indirect doctest
Modular Symbols space of dimension 8 for Gamma_1(7) of weight 3 with sign 0 and over Rational Field
"""
return ("Modular Symbols space of dimension %s for Gamma_1(%s) of weight %s with sign %s " + \
"and over %s")%(self.dimension(), self.level(),self.weight(),
self.sign(), self.base_ring())
def _cuspidal_submodule_dimension_formula(self):
r"""
Return the dimension of the cuspidal subspace, using the formula.
EXAMPLES::
sage: M = ModularSymbols(Gamma1(11),4)
sage: M.cuspidal_subspace().dimension()
20
sage: M._cuspidal_submodule_dimension_formula()
20
"""
if self.sign() == 0:
m = 2
else:
m = 1
return m * self.group().dimension_cusp_forms(self.weight())
def _cuspidal_new_submodule_dimension_formula(self):
r"""
Return the dimension of the new cuspidal subspace, via the formula.
EXAMPLES:
sage: M = ModularSymbols(Gamma1(22),2)
sage: M._cuspidal_new_submodule_dimension_formula()
8
sage: M.cuspidal_subspace().new_subspace().dimension()
8
"""
if self.sign() == 0:
m = 2
else:
m = 1
return m * self.group().dimension_new_cusp_forms(self.weight())
def _compute_hecke_matrix_prime_power(self, p, r):
r"""
Compute and return the matrix of the Hecke operator `T(p^r)`.
EXAMPLES::
sage: m = ModularSymbols(Gamma1(11),2)
sage: m._compute_hecke_matrix_prime_power(3,4).charpoly('x')
x^11 - 291*x^10 + 30555*x^9 - 1636145*x^8 + 59637480*x^7 + 1983040928*x^6 - 401988683888*x^5 - 14142158875680*x^4 + 3243232720819520*x^3 - 103658398669404480*x^2 + 197645665452381696*x - 97215957397309696
"""
return self._compute_hecke_matrix_prime(p**r)
def _degeneracy_raising_matrix_1(self, M):
r"""
Return the matrix of the degeneracy raising map to `M`.
INPUT:
- ``M`` -- an ambient space of Gamma1 modular symbols, of level a
multiple of the level of self
OUTPUT:
(matrix) The matrix of the degeneracy raising matrix to the higher level.
EXAMPLES::
sage: M = ModularSymbols(Gamma1(7),3)
sage: N = ModularSymbols(Gamma1(21), 3)
sage: M._degeneracy_raising_matrix_1(N)
8 x 64 dense matrix over Rational Field
sage: M.dimension()
8
sage: N.dimension()
64
"""
level = int(M.level())
N = self.level()
H = arithgroup.degeneracy_coset_representatives_gamma1(M.level(), N, 1)
MS = matrix_space.MatrixSpace(self.base_ring(), self.dimension(), M.dimension())
if self.dimension() == 0 or M.dimension() == 0:
return MS(0)
rows = []
B = self.manin_basis()
syms = self.manin_symbols()
k = self.weight()
G = matrix_space.MatrixSpace(integer_ring.IntegerRing(),2)
H = [G(h) for h in H]
for n in B:
z = M(0)
s = syms.manin_symbol(n)
g = G(list(s.lift_to_sl2z(N)))
i = s.i
for h in H:
hg = h*g
z += M((i, hg[1,0], hg[1,1]))
rows.append(z.element())
A = MS(rows)
return A
def boundary_space(self):
r"""
Return the space of boundary modular symbols for this space.
EXAMPLES::
sage: M = ModularSymbols(100,2)
sage: M.boundary_space()
Space of Boundary Modular Symbols for Congruence Subgroup Gamma0(100) of weight 2 and over Rational Field
"""
try:
return self.__boundary_space
except AttributeError:
pass
self.__boundary_space = boundary.BoundarySpace_wtk_g1(
self.level(), self.weight(), self.sign(), self.base_ring())
return self.__boundary_space
def manin_symbols(self):
r"""
Return the Manin symbol list of this modular symbol space.
EXAMPLES::
sage: M = ModularSymbols(Gamma1(30),4)
sage: M.manin_symbols()
Manin Symbol List of weight 4 for Gamma1(30)
sage: len(M.manin_symbols())
1728
"""
try:
return self.__manin_symbols
except AttributeError:
self.__manin_symbols = manin_symbols.ManinSymbolList_gamma1(
level=self.level(), weight=self.weight())
return self.__manin_symbols
def modular_symbols_of_level(self, N):
r"""
Returns a space of modular symbols with the same parameters as
this space except with level `N`.
INPUT:
- ``N`` (int) -- a positive integer.
OUTPUT:
(Modular Symbol space) A space of modular symbols with the
same defining properties (weight, sign, etc.) as this space
except with level `N`.
For example, if self is the space of modular symbols of weight `2`
for `\Gamma_0(22)`, and level is `11`, then this function returns
the modular symbol space of weight `2` for `\Gamma_0(11)`.
EXAMPLES::
sage: M = ModularSymbols(Gamma1(30),4); M
Modular Symbols space of dimension 144 for Gamma_1(30) of weight 4 with sign 0 and over Rational Field
sage: M.modular_symbols_of_level(22)
Modular Symbols space of dimension 90 for Gamma_1(22) of weight 4 with sign 0 and over Rational Field
"""
return modsym.ModularSymbols(arithgroup.Gamma1(N), self.weight(),self.sign(), self.base_ring())
class ModularSymbolsAmbient_wtk_gamma_h(ModularSymbolsAmbient):
def __init__(self, group, weight, sign, F, custom_init=None):
r"""
Initialize a space of modular symbols for `\Gamma_H(N)`.
INPUT:
- ``group`` - a congruence subgroup
`\Gamma_H(N)`.
- ``weight`` - int, the weight = 2
- ``sign`` - int, either -1, 0, or 1
- ``F`` - field
EXAMPLES::
sage: ModularSymbols(GammaH(15,[4]),2)
Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(15) with H generated by [4] of weight 2 with sign 0 and over Rational Field
"""
ModularSymbolsAmbient.__init__(self,
weight=weight, group=group,
sign=sign, base_ring=F, custom_init=custom_init)
def _dimension_formula(self):
r"""
Return None: we have no dimension formulas for `\Gamma_H(N)` spaces.
EXAMPLES::
sage: M = ModularSymbols(GammaH(15,[4]),2)
sage: M.dimension()
9
sage: M._dimension_formula()
"""
return None
def _repr_(self):
r"""
Return a string representation of this space.
EXAMPLES::
sage: M = ModularSymbols(GammaH(15,[4]),2)
sage: M # indirect doctest
Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(15) with H generated by [4] of weight 2 with sign 0 and over Rational Field
"""
return ("Modular Symbols space of dimension %s for %s of weight %s with sign %s " + \
"and over %s")%(self.dimension(), self.group(),self.weight(),
self.sign(), self.base_ring())
def _cuspidal_submodule_dimension_formula(self):
r"""
Return None: we have no dimension formulas for `\Gamma_H(N)` spaces.
EXAMPLES::
sage: ModularSymbols(GammaH(15,[4]),2)._cuspidal_submodule_dimension_formula() is None
True
"""
return None
def _cuspidal_new_submodule_dimension_formula(self):
r"""
Return None: we have no dimension formulas for `\Gamma_H(N)` spaces.
EXAMPLES::
sage: ModularSymbols(GammaH(15,[4]),2)._cuspidal_new_submodule_dimension_formula() is None
True
"""
return None
def _compute_hecke_matrix_prime_power(self, p, r):
r"""
Return matrix of a prime-power Hecke operator.
EXAMPLES::
sage: M = ModularSymbols(GammaH(15,[4]),2)
sage: M._compute_hecke_matrix_prime_power(2, 3)
[10 0 5 0 1 0 0 4 0]
[ 0 10 0 0 -4 -5 0 -1 6]
[ 5 0 10 0 -4 0 0 -1 0]
[ 0 0 0 5 -7 0 10 -3 4]
[ 0 0 0 0 -1 0 0 -4 0]
[ 0 -5 0 0 -1 10 0 -4 -6]
[ 0 0 0 10 -3 0 5 -7 -4]
[ 0 0 0 0 -4 0 0 -1 0]
[ 0 0 0 0 0 0 0 0 3]
sage: M.hecke_matrix(7)^2 == M.hecke_matrix(49) + 7 * M.diamond_bracket_operator(7).matrix() # indirect doctest
True
"""
return self._compute_hecke_matrix_prime(p**r)
def _degeneracy_raising_matrix_1(self, level):
r"""
Return matrix of a degeneracy raising map.
EXAMPLES::
sage: ModularSymbols(GammaH(15,[4]),2)._degeneracy_raising_matrix_1(ModularSymbols(GammaH(30, [19]), 2))
Traceback (most recent call last):
...
NotImplementedError
"""
raise NotImplementedError
def boundary_space(self):
r"""
Return the space of boundary modular symbols for this space.
EXAMPLES::
sage: M = ModularSymbols(GammaH(15,[4]),2)
sage: M.boundary_space()
Boundary Modular Symbols space for Congruence Subgroup Gamma_H(15) with H generated by [4] of weight 2 over Rational Field
"""
try:
return self.__boundary_space
except AttributeError:
pass
self.__boundary_space = boundary.BoundarySpace_wtk_gamma_h(
self.group(), self.weight(), self.sign(), self.base_ring())
return self.__boundary_space
def manin_symbols(self):
r"""
Return the Manin symbol list of this modular symbol space.
EXAMPLES::
sage: M = ModularSymbols(GammaH(15,[4]),2)
sage: M.manin_symbols()
Manin Symbol List of weight 2 for Congruence Subgroup Gamma_H(15) with H generated by [4]
sage: len(M.manin_symbols())
96
"""
try:
return self.__manin_symbols
except AttributeError:
self.__manin_symbols = manin_symbols.ManinSymbolList_gamma_h(
group=self.group(), weight=self.weight())
return self.__manin_symbols
def modular_symbols_of_level(self, N):
r"""
Returns a space of modular symbols with the same parameters as
this space except with level `N`, which should be either a divisor or a
multiple of the level of self.
TESTS::
sage: M = ModularSymbols(GammaH(15,[7]),6)
sage: M.modular_symbols_of_level(5)
Modular Symbols space of dimension 4 for Gamma_0(5) of weight 6 with sign 0 over Rational Field
sage: M.modular_symbols_of_level(30)
Traceback (most recent call last):
...
ValueError: N (=30) should be a factor of the level of this space (=15)
sage: M.modular_symbols_of_level(73)
Traceback (most recent call last):
...
ValueError: N (=73) should be a factor of the level of this space (=15)
"""
if self.level() % N == 0:
return modsym.ModularSymbols(self.group().restrict(N), self.weight(), self.sign(), self.base_ring())
else:
raise ValueError, "N (=%s) should be a factor of the level of this space (=%s)" % (N, self.level())
class ModularSymbolsAmbient_wtk_eps(ModularSymbolsAmbient):
def __init__(self, eps, weight, sign, base_ring, custom_init=None):
"""
Space of modular symbols with given weight, character, base ring and
sign.
INPUT:
- ``eps`` - dirichlet.DirichletCharacter, the
"Nebentypus" character.
- ``weight`` - int, the weight = 2
- ``sign`` - int, either -1, 0, or 1
- ``base_ring`` - the base ring. It must be possible to change the ring
of the character to this base ring (not always canonically).
EXAMPLES::
sage: eps = DirichletGroup(4).gen(0)
sage: eps.order()
2
sage: ModularSymbols(eps, 2)
Modular Symbols space of dimension 0 and level 4, weight 2, character [-1], sign 0, over Rational Field
sage: ModularSymbols(eps, 3)
Modular Symbols space of dimension 2 and level 4, weight 3, character [-1], sign 0, over Rational Field
We next create a space with character of order bigger than 2.
::
sage: eps = DirichletGroup(5).gen(0)
sage: eps # has order 4
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4
sage: ModularSymbols(eps, 2).dimension()
0
sage: ModularSymbols(eps, 3).dimension()
2
Here is another example::
sage: G, e = DirichletGroup(5).objgen()
sage: M = ModularSymbols(e,3)
sage: loads(M.dumps()) == M
True
"""
level = eps.modulus()
ModularSymbolsAmbient.__init__(self,
weight = weight,
group = arithgroup.Gamma1(level),
sign = sign,
base_ring = base_ring,
character = eps.change_ring(base_ring),
custom_init=custom_init)
def _repr_(self):
r"""
Return a string representation of this space.
EXAMPLES::
sage: G, e = DirichletGroup(5).objgen()
sage: M = ModularSymbols(e,3)
sage: M # indirect doctest
Modular Symbols space of dimension 2 and level 5, weight 3, character [zeta4], sign 0, over Cyclotomic Field of order 4 and degree 2
"""
return ("Modular Symbols space of dimension %s and level %s, weight %s, character %s, sign %s, " + \
"over %s")%(self.dimension(), self.level(), self.weight(),
self.character()._repr_short_(), self.sign(), self.base_ring())
def _cuspidal_submodule_dimension_formula(self):
r"""
Return the dimension for the cuspidal subspace of this space, given by the formula.
EXAMPLES::
sage: G, e = DirichletGroup(50).objgen()
sage: M = ModularSymbols(e^2,2)
sage: M.dimension()
16
sage: M._cuspidal_submodule_dimension_formula()
12
"""
if self.base_ring().characteristic() != 0:
raise NotImplementedError
if self.sign() == 0:
m = 2
else:
m = 1
return m * self.group().dimension_cusp_forms(self.weight(), eps=self.character())
def _cuspidal_new_submodule_dimension_formula(self):
r"""
Return the dimension for the new cuspidal subspace of this space, given by the formula.
EXAMPLES::
sage: G, e = DirichletGroup(50).objgen()
sage: M = ModularSymbols(e,3)
sage: M.dimension()
30
sage: M._cuspidal_new_submodule_dimension_formula()
10
"""
if self.base_ring().characteristic() != 0:
raise NotImplementedError
if self.sign() == 0:
m = 2
else:
m = 1
return m * self.group().dimension_new_cusp_forms(self.weight(), eps=self.character())
def _matrix_of_operator_on_modular_symbols(self, codomain, R, character_twist=False):
"""
INPUT:
- ``self`` - this space of modular symbols
- ``codomain`` - space of modular symbols
- ``R`` - list of lists [a,b,c,d] of length 4, which
we view as elements of GL_2(Q).
OUTPUT: a matrix, which represents the operator
.. math::
x \mapsto \sum_{g in R} g.x
where g.x is the formal linear fractional transformation on modular
symbols.
EXAMPLES::
sage: G, e = DirichletGroup(5).objgen()
sage: M = ModularSymbols(e,3)
sage: M.dimension()
2
sage: M._matrix_of_operator_on_modular_symbols(M,HeilbronnCremona(3))
[ 6 6]
[ 0 10]
"""
eps = self.character()
rows = []
for b in self.basis():
v = formal_sum.FormalSum(0, check=False)
for c, x in b.modular_symbol_rep():
for g in R:
y = x.apply(g)
if character_twist:
v += y*c*eps(g[0])
else:
v += y*c
w = codomain(v).element()
rows.append(w)
M = matrix_space.MatrixSpace(self.base_ring(), len(rows), codomain.degree(), sparse=False)
return M(rows)
def _degeneracy_raising_matrix_1(self, M):
r"""
Return the matrix of the degeneracy raising map to ``M``, which should
be a space of modular symbols with level a multiple of the level of
self and with compatible character.
INPUT:
- ``M`` -- a space of modular symbols with character, whose level
should be an integer multiple of the level of self, and whose
character should be the Dirichlet character at that level obtained by
extending the character of self.
The input is *not* sanity-checked in any way -- use with care!
OUTPUT:
(matrix) The matrix of the degeneracy raising matrix to the higher level.
EXAMPLES::
sage: eps = DirichletGroup(4).gen(0)
sage: M = ModularSymbols(eps, 3); M
Modular Symbols space of dimension 2 and level 4, weight 3, character [-1], sign 0, over Rational Field
sage: M._degeneracy_raising_matrix_1(ModularSymbols(eps.extend(20), 3))
[ 1 0 0 0 -1 -1 3 1 0 2 -3 0]
[ 0 5 1 -2 -3 3 0 4 -1 5 -7 -1]
"""
level = int(M.level())
N = self.level()
H = arithgroup.degeneracy_coset_representatives_gamma0(M.level(), N, 1)
MS = matrix_space.MatrixSpace(self.base_ring(), self.dimension(), M.dimension())
if self.dimension() == 0 or M.dimension() == 0:
return MS(0)
rows = []
B = self.manin_basis()
syms = self.manin_symbols()
k = self.weight()
G = matrix_space.MatrixSpace(integer_ring.IntegerRing(),2)
H = [G(h) for h in H]
eps = self.character()
for n in B:
z = M(0)
s = syms.manin_symbol(n)
g = G(list(s.lift_to_sl2z(N)))
i = s.i
for h in H:
hg = h*g
z += eps(h[0,0])*M((i, hg[1,0], hg[1,1]))
rows.append(z.element())
A = MS(rows)
return A
def _dimension_formula(self):
r"""
Return None: we have no dimension formula for `\Gamma_H(N)` spaces.
EXAMPLES::
sage: eps = DirichletGroup(5).gen(0)
sage: M = ModularSymbols(eps, 2)
sage: M.dimension()
0
sage: M._dimension_formula()
"""
return None
def boundary_space(self):
r"""
Return the space of boundary modular symbols for this space.
EXAMPLES::
sage: eps = DirichletGroup(5).gen(0)
sage: M = ModularSymbols(eps, 2)
sage: M.boundary_space()
Boundary Modular Symbols space of level 5, weight 2, character [zeta4] and dimension 0 over Cyclotomic Field of order 4 and degree 2
"""
try:
return self.__boundary_space
except AttributeError:
pass
self.__boundary_space = boundary.BoundarySpace_wtk_eps(
self.character(), self.weight(), self.sign())
return self.__boundary_space
def manin_symbols(self):
r"""
Return the Manin symbol list of this modular symbol space.
EXAMPLES::
sage: eps = DirichletGroup(5).gen(0)
sage: M = ModularSymbols(eps, 2)
sage: M.manin_symbols()
Manin Symbol List of weight 2 for Gamma1(5) with character [zeta4]
sage: len(M.manin_symbols())
6
"""
try:
return self.__manin_symbols
except AttributeError:
self.__manin_symbols = manin_symbols.ManinSymbolList_character(
character=self.character(), weight=self.weight())
return self.__manin_symbols
def modular_symbols_of_level(self, N):
r"""
Returns a space of modular symbols with the same parameters as
this space except with level `N`.
INPUT:
- ``N`` (int) -- a positive integer.
OUTPUT:
(Modular Symbol space) A space of modular symbols with the
same defining properties (weight, sign, etc.) as this space
except with level `N`.
EXAMPLES::
sage: eps = DirichletGroup(5).gen(0)
sage: M = ModularSymbols(eps, 2); M
Modular Symbols space of dimension 0 and level 5, weight 2, character [zeta4], sign 0, over Cyclotomic Field of order 4 and degree 2
sage: M.modular_symbols_of_level(15)
Modular Symbols space of dimension 0 and level 15, weight 2, character [1, zeta4], sign 0, over Cyclotomic Field of order 4 and degree 2
"""
if self.level() % N == 0:
eps = self.character().restrict(N)
elif N % self.level() == 0:
eps = self.character().extend(N)
else:
raise ValueError, "The level N (=%s) must be a divisor or multiple of the modulus of the character (=%s)"%(N, self.level())
return modsym.ModularSymbols(eps, self.weight(), self.sign(), self.base_ring())
def modular_symbols_of_sign(self, sign):
r"""
Returns a space of modular symbols with the same defining
properties (weight, level, etc.) as this space except with given
sign.
INPUT:
- ``sign`` (int) -- A sign (`+1`, `-1` or `0`).
OUTPUT:
(ModularSymbolsAmbient) A space of modular symbols with the
same defining properties (weight, level, etc.) as this space
except with given sign.
EXAMPLES::
sage: eps = DirichletGroup(5).gen(0)
sage: M = ModularSymbols(eps, 2); M
Modular Symbols space of dimension 0 and level 5, weight 2, character [zeta4], sign 0, over Cyclotomic Field of order 4 and degree 2
sage: M.modular_symbols_of_sign(0) == M
True
sage: M.modular_symbols_of_sign(+1)
Modular Symbols space of dimension 0 and level 5, weight 2, character [zeta4], sign 1, over Cyclotomic Field of order 4 and degree 2
sage: M.modular_symbols_of_sign(-1)
Modular Symbols space of dimension 0 and level 5, weight 2, character [zeta4], sign -1, over Cyclotomic Field of order 4 and degree 2
"""
return modsym.ModularSymbols(self.character(), self.weight(), sign, self.base_ring())
def modular_symbols_of_weight(self, k):
r"""
Returns a space of modular symbols with the same defining
properties (weight, sign, etc.) as this space except with weight
`k`.
INPUT:
- ``k`` (int) -- A positive integer.
OUTPUT:
(ModularSymbolsAmbient) A space of modular symbols with the
same defining properties (level, sign) as this space
except with given weight.
EXAMPLES::
sage: eps = DirichletGroup(5).gen(0)
sage: M = ModularSymbols(eps, 2); M
Modular Symbols space of dimension 0 and level 5, weight 2, character [zeta4], sign 0, over Cyclotomic Field of order 4 and degree 2
sage: M.modular_symbols_of_weight(3)
Modular Symbols space of dimension 2 and level 5, weight 3, character [zeta4], sign 0, over Cyclotomic Field of order 4 and degree 2
sage: M.modular_symbols_of_weight(2) == M
True
"""
return modsym.ModularSymbols(self.character(), k, self.sign(), self.base_ring())
def _hecke_images(self, i, v):
"""
Return images of the `i`-th standard basis vector under the
Hecke operators `T_p` for all integers in `v`.
INPUT:
- ``i`` - nonnegative integer
- ``v`` - a list of positive integer
OUTPUT:
- ``matrix`` - whose rows are the Hecke images
EXAMPLES::
sage: G, e = DirichletGroup(50,QQ).objgen()
sage: M = ModularSymbols(e^2,2)
sage: M.dimension()
15
sage: M._hecke_images(8,range(1,5))
[ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0]
[ 0 1 0 2 0 -1 1 1 0 0 0 0 0 0 0]
[ 0 1 1 -1 -1 0 -1 1 1 0 1 2 0 -2 2]
"""
if self.weight() != 2:
raise NotImplementedError, "hecke images only implemented when the weight is 2"
chi = self.character()
c = self.manin_generators()[self.manin_basis()[i]]
N = self.level()
if chi.order() > 2:
return heilbronn.hecke_images_nonquad_character_weight2(c.u,c.v,N,
v, chi, self.manin_gens_to_basis())
else:
return heilbronn.hecke_images_quad_character_weight2(c.u,c.v,N,
v, chi, self.manin_gens_to_basis())
raise NotImplementedError
