constructor.py

# -*- coding: utf-8 -*-
"""
Creating Spaces of Modular Forms

EXAMPLES::

    sage: m = ModularForms(Gamma1(4),11)
    sage: m
    Modular Forms space of dimension 6 for Congruence Subgroup Gamma1(4) of weight 11 over Rational Field
    sage: m.basis()
    [
    q - 134*q^5 + O(q^6),
    q^2 + 80*q^5 + O(q^6),
    q^3 + 16*q^5 + O(q^6),
    q^4 - 4*q^5 + O(q^6),
    1 + 4092/50521*q^2 + 472384/50521*q^3 + 4194300/50521*q^4 + O(q^6),
    q + 1024*q^2 + 59048*q^3 + 1048576*q^4 + 9765626*q^5 + O(q^6)
    ]

"""

#########################################################################
#       Copyright (C) 2004--2006 William Stein <[email protected]>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#                  http://www.gnu.org/licenses/
#########################################################################

import weakref
import re

import sage.modular.arithgroup.all as arithgroup
import sage.modular.dirichlet as dirichlet
import sage.rings.all as rings

from sage.rings.commutative_ring import is_CommutativeRing

import ambient_eps
import ambient_g0
import ambient_g1
import ambient_R
import defaults


def canonical_parameters(group, level, weight, base_ring):
    """
    Given a group, level, weight, and base_ring as input by the user,
    return a canonicalized version of them, where level is a Sage
    integer, group really is a group, weight is a Sage integer, and
    base_ring a Sage ring. Note that we can't just get the level from
    the group, because we have the convention that the character for
    Gamma1(N) is None (which makes good sense).

    INPUT:


    -  ``group`` - int, long, Sage integer, group,
       dirichlet character, or

    -  ``level`` - int, long, Sage integer, or group

    -  ``weight`` - coercible to Sage integer

    -  ``base_ring`` - commutative Sage ring


    OUTPUT:


    -  ``level`` - Sage integer

    -  ``group`` - congruence subgroup

    -  ``weight`` - Sage integer

    -  ``ring`` - commutative Sage ring


    EXAMPLES::

        sage: from sage.modular.modform.constructor import canonical_parameters
        sage: v = canonical_parameters(5, 5, int(7), ZZ); v
        (5, Congruence Subgroup Gamma0(5), 7, Integer Ring)
        sage: type(v[0]), type(v[1]), type(v[2]), type(v[3])
        (<type 'sage.rings.integer.Integer'>,
         <class 'sage.modular.arithgroup.congroup_gamma0.Gamma0_class_with_category'>,
         <type 'sage.rings.integer.Integer'>,
         <type 'sage.rings.integer_ring.IntegerRing_class'>)
        sage: canonical_parameters( 5, 7, 7, ZZ )
        Traceback (most recent call last):
        ...
        ValueError: group and level do not match.
    """
    weight = rings.Integer(weight)
    if weight <= 0:
        raise NotImplementedError, "weight must be at least 1"

    if isinstance(group, dirichlet.DirichletCharacter):
        if ( group.level() != rings.Integer(level) ):
            raise ValueError, "group.level() and level do not match."
        group = group.minimize_base_ring()
        level = rings.Integer(level)

    elif arithgroup.is_CongruenceSubgroup(group):
        if ( rings.Integer(level) != group.level() ):
            raise ValueError, "group.level() and level do not match."
        # normalize the case of SL2Z
        if arithgroup.is_SL2Z(group) or \
           arithgroup.is_Gamma1(group) and group.level() == rings.Integer(1):
            group = arithgroup.Gamma0(rings.Integer(1))

    elif group is None:
        pass

    else:
        try:
            m = rings.Integer(group)
        except TypeError:
            raise TypeError, "group of unknown type."
        level = rings.Integer(level)
        if ( m != level ):
            raise ValueError, "group and level do not match."
        group = arithgroup.Gamma0(m)

    if not is_CommutativeRing(base_ring):
        raise TypeError, "base_ring (=%s) must be a commutative ring"%base_ring

    # it is *very* important to include the level as part of the data
    # that defines the key, since dirichlet characters of different
    # levels can compare equal, but define much different modular
    # forms spaces.
    return level, group, weight, base_ring

_cache = {}

def ModularForms_clear_cache():
    """
    Clear the cache of modular forms.

    EXAMPLES::

        sage: M = ModularForms(37,2)
        sage: sage.modular.modform.constructor._cache == {}
        False

    ::

        sage: sage.modular.modform.constructor.ModularForms_clear_cache()
        sage: sage.modular.modform.constructor._cache
        {}
    """
    global _cache
    _cache = {}

def ModularForms(group  = 1,
                 weight = 2,
                 base_ring = None,
                 use_cache = True,
                 prec = defaults.DEFAULT_PRECISION):
    r"""
    Create an ambient space of modular forms.

    INPUT:


    -  ``group`` - A congruence subgroup or a Dirichlet
       character eps.

    -  ``weight`` - int, the weight, which must be an
       integer = 1.

    -  ``base_ring`` - the base ring (ignored if group is
       a Dirichlet character)


    Create using the command ModularForms(group, weight, base_ring)
    where group could be either a congruence subgroup or a Dirichlet
    character.

    EXAMPLES: First we create some spaces with trivial character::

        sage: ModularForms(Gamma0(11),2).dimension()
        2
        sage: ModularForms(Gamma0(1),12).dimension()
        2

    If we give an integer N for the congruence subgroup, it defaults to
    `\Gamma_0(N)`::

        sage: ModularForms(1,12).dimension()
        2
        sage: ModularForms(11,4)
        Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(11) of weight 4 over Rational Field

    We create some spaces for `\Gamma_1(N)`.

    ::

        sage: ModularForms(Gamma1(13),2)
        Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
        sage: ModularForms(Gamma1(13),2).dimension()
        13
        sage: [ModularForms(Gamma1(7),k).dimension() for k in [2,3,4,5]]
        [5, 7, 9, 11]
        sage: ModularForms(Gamma1(5),11).dimension()
        12

    We create a space with character::

        sage: e = (DirichletGroup(13).0)^2
        sage: e.order()
        6
        sage: M = ModularForms(e, 2); M
        Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2
        sage: f = M.T(2).charpoly('x'); f
        x^3 + (-2*zeta6 - 2)*x^2 - 2*zeta6*x + 14*zeta6 - 7
        sage: f.factor()
        (x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)

    We can also create spaces corresponding to the groups `\Gamma_H(N)` intermediate
    between `\Gamma_0(N)` and `\Gamma_1(N)`::

        sage: G = GammaH(30, [11])
        sage: M = ModularForms(G, 2); M
        Modular Forms space of dimension 20 for Congruence Subgroup Gamma_H(30) with H generated by [11] of weight 2 over Rational Field
        sage: M.T(7).charpoly().factor()  # long time (7s on sage.math, 2011)
        (x + 4) * x^2 * (x - 6)^4 * (x + 6)^4 * (x - 8)^7 * (x^2 + 4)

    More examples of spaces with character::

        sage: e = DirichletGroup(5, RationalField()).gen(); e
        Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1

        sage: m = ModularForms(e, 2); m
        Modular Forms space of dimension 2, character [-1] and weight 2 over Rational Field
        sage: m == loads(dumps(m))
        True
        sage: m.T(2).charpoly('x')
        x^2 - 1
        sage: m = ModularForms(e, 6); m.dimension()
        4
        sage: m.T(2).charpoly('x')
        x^4 - 917*x^2 - 42284

    This came up in a subtle bug (trac #5923)::

        sage: ModularForms(gp(1), gap(12))
        Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field

    This came up in another bug (related to trac #8630)::

        sage: chi = DirichletGroup(109, CyclotomicField(3)).0
        sage: ModularForms(chi, 2, base_ring = CyclotomicField(15))
        Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2 over Cyclotomic Field of order 15 and degree 8

    We create some weight 1 spaces. The first example works fine, since we can prove purely by Riemann surface theory that there are no weight 1 cusp forms::

        sage: M = ModularForms(Gamma1(11), 1); M
        Modular Forms space of dimension 5 for Congruence Subgroup Gamma1(11) of weight 1 over Rational Field
        sage: M.basis()
        [
        1 + 22*q^5 + O(q^6),
        q + 4*q^5 + O(q^6),
        q^2 - 4*q^5 + O(q^6),
        q^3 - 5*q^5 + O(q^6),
        q^4 - 3*q^5 + O(q^6)
        ]
        sage: M.cuspidal_subspace().basis()
        [
        ]
        sage: M == M.eisenstein_subspace()
        True

    This example doesn't work so well, because we can't calculate the cusp
    forms; but we can still work with the Eisenstein series.

        sage: M = ModularForms(Gamma1(57), 1); M
        Modular Forms space of dimension (unknown) for Congruence Subgroup Gamma1(57) of weight 1 over Rational Field
        sage: M.basis()
        Traceback (most recent call last):
        ...
        NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general
        sage: M.cuspidal_subspace().basis()
        Traceback (most recent call last):
        ...
        NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general

        sage: E = M.eisenstein_subspace(); E
        Eisenstein subspace of dimension 36 of Modular Forms space of dimension (unknown) for Congruence Subgroup Gamma1(57) of weight 1 over Rational Field
        sage: (E.0 + E.2).q_expansion(40)
        1 + q^2 + 1473/2*q^36 - 1101/2*q^37 + q^38 - 373/2*q^39 + O(q^40)

    """
    if isinstance(group, dirichlet.DirichletCharacter):
        if base_ring is None:
            base_ring = group.minimize_base_ring().base_ring()
    if base_ring is None:
        base_ring = rings.QQ

    if isinstance(group, dirichlet.DirichletCharacter) \
           or arithgroup.is_CongruenceSubgroup(group):
        level = group.level()
    else:
        level = group

    key = canonical_parameters(group, level, weight, base_ring)

    if use_cache and _cache.has_key(key):
         M = _cache[key]()
         if not (M is None):
             M.set_precision(prec)
             return M

    (level, group, weight, base_ring) = key

    M = None
    if arithgroup.is_Gamma0(group):
        M = ambient_g0.ModularFormsAmbient_g0_Q(group.level(), weight)
        if base_ring != rings.QQ:
            M = ambient_R.ModularFormsAmbient_R(M, base_ring)

    elif arithgroup.is_Gamma1(group):
        M = ambient_g1.ModularFormsAmbient_g1_Q(group.level(), weight)
        if base_ring != rings.QQ:
            M = ambient_R.ModularFormsAmbient_R(M, base_ring)

    elif arithgroup.is_GammaH(group):
        M = ambient_g1.ModularFormsAmbient_gH_Q(group, weight)
        if base_ring != rings.QQ:
            M = ambient_R.ModularFormsAmbient_R(M, base_ring)

    elif isinstance(group, dirichlet.DirichletCharacter):
        eps = group
        if eps.base_ring().characteristic() != 0:
            # TODO -- implement this
            # Need to add a lift_to_char_0 function for characters,
            # and need to still remember eps.
            raise NotImplementedError, "currently the character must be over a ring of characteristic 0."
        eps = eps.minimize_base_ring()
        if eps.is_trivial():
            return ModularForms(eps.modulus(), weight, base_ring,
                                use_cache = use_cache,
                                prec = prec)
        M = ambient_eps.ModularFormsAmbient_eps(eps, weight)
        if base_ring != eps.base_ring():
            M = M.base_extend(base_ring) # ambient_R.ModularFormsAmbient_R(M, base_ring)

    if M is None:
        raise NotImplementedError, \
           "computation of requested space of modular forms not defined or implemented"

    M.set_precision(prec)
    _cache[key] = weakref.ref(M)
    return M


def CuspForms(group  = 1,
              weight = 2,
              base_ring = None,
              use_cache = True,
              prec = defaults.DEFAULT_PRECISION):
    """
    Create a space of cuspidal modular forms.

    See the documentation for the ModularForms command for a
    description of the input parameters.

    EXAMPLES::

        sage: CuspForms(11,2)
        Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
    """
    return ModularForms(group, weight, base_ring,
                        use_cache=use_cache, prec=prec).cuspidal_submodule()


def EisensteinForms(group  = 1,
              weight = 2,
              base_ring = None,
              use_cache = True,
              prec = defaults.DEFAULT_PRECISION):
    """
    Create a space of eisenstein modular forms.

    See the documentation for the ModularForms command for a
    description of the input parameters.

    EXAMPLES::

        sage: EisensteinForms(11,2)
        Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
    """
    return ModularForms(group, weight, base_ring,
                        use_cache=use_cache, prec=prec).eisenstein_submodule()



def Newforms(group, weight=2, base_ring=None, names=None):
    r"""
    Returns a list of the newforms of the given weight and level (or weight,
    level and character). These are calculated as
    `\operatorname{Gal}(\overline{F} / F)`-orbits, where `F` is the given base
    field.

    INPUT:


    -  ``group`` - the congruence subgroup of the newform, or a Nebentypus
       character

    -  ``weight`` - the weight of the newform (default 2)

    -  ``base_ring`` - the base ring (defaults to `\QQ` for spaces without
       character, or the base ring of the character otherwise)

    -  ``names`` - if the newform has coefficients in a
       number field, a generator name must be specified


    EXAMPLES::

        sage: Newforms(11, 2)
        [q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)]
        sage: Newforms(65, names='a')
        [q - q^2 - 2*q^3 - q^4 - q^5 + O(q^6),
         q + a1*q^2 + (a1 + 1)*q^3 + (-2*a1 - 1)*q^4 + q^5 + O(q^6),
         q + a2*q^2 + (-a2 + 1)*q^3 + q^4 - q^5 + O(q^6)]

    A more complicated example involving both a nontrivial character, and a
    base field that is not minimal for that character::

        sage: K.<i> = QuadraticField(-1)
        sage: chi = DirichletGroup(5, K)[3]
        sage: len(Newforms(chi, 7, names='a'))
        1
        sage: x = polygen(K); L.<c> = K.extension(x^2 - 402*i)
        sage: N = Newforms(chi, 7, base_ring = L); len(N)
        2
        sage: sorted([N[0][2], N[1][2]]) == sorted([1/2*c - 5/2*i - 5/2, -1/2*c - 5/2*i - 5/2])
        True

    We test that :trac:`8630` is fixed::

        sage: chi = DirichletGroup(109, CyclotomicField(3)).0
        sage: CuspForms(chi, 2, base_ring = CyclotomicField(9))
        Cuspidal subspace of dimension 8 of Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2 over Cyclotomic Field of order 9 and degree 6
    """
    return CuspForms(group, weight, base_ring).newforms(names)


def Newform(identifier, group=None, weight=2, base_ring=rings.QQ, names=None):
    """
    INPUT:


    -  ``identifier`` - a canonical label, or the index of
       the specific newform desired

    -  ``group`` - the congruence subgroup of the newform

    -  ``weight`` - the weight of the newform (default 2)

    -  ``base_ring`` - the base ring

    -  ``names`` - if the newform has coefficients in a
       number field, a generator name must be specified


    EXAMPLES::

        sage: Newform('67a', names='a')
        q + 2*q^2 - 2*q^3 + 2*q^4 + 2*q^5 + O(q^6)
        sage: Newform('67b', names='a')
        q + a1*q^2 + (-a1 - 3)*q^3 + (-3*a1 - 3)*q^4 - 3*q^5 + O(q^6)
    """
    if isinstance(group, str) and names is None:
        names = group
    if isinstance(identifier, str):
        group, identifier = parse_label(identifier)
        if weight != 2:
            raise ValueError, "Canonical label not implemented for higher weight forms."
        elif base_ring != rings.QQ:
            raise ValueError, "Canonical label not implemented except for over Q."
    elif group is None:
        raise ValueError, "Must specify a group or a label."
    return Newforms(group, weight, base_ring, names=names)[identifier]


def parse_label(s):
    """
    Given a string s corresponding to a newform label, return the
    corresponding group and index.

    EXAMPLES::

        sage: sage.modular.modform.constructor.parse_label('11a')
        (Congruence Subgroup Gamma0(11), 0)
        sage: sage.modular.modform.constructor.parse_label('11aG1')
        (Congruence Subgroup Gamma1(11), 0)
        sage: sage.modular.modform.constructor.parse_label('11wG1')
        (Congruence Subgroup Gamma1(11), 22)
    """
    m = re.match(r'(\d+)([a-z]+)((?:G.*)?)$', s)
    if not m:
        raise ValueError, "Invalid label: %s" % s
    N, order, G = m.groups()
    N = int(N)
    index = 0
    for c in reversed(order):
        index = 26*index + ord(c)-ord('a')
    if G == '' or G == 'G0':
        G = arithgroup.Gamma0(N)
    elif G == 'G1':
        G = arithgroup.Gamma1(N)
    elif G[:2] == 'GH':
        if G[2] != '[' or G[-1] != ']':
            raise ValueError, "Invalid congruence subgroup label: %s" % G
        gens = [int(g.strip()) for g in G[3:-1].split(',')]
        return arithgroup.GammaH(N, gens)
    else:
        raise ValueError, "Invalid congruence subgroup label: %s" % G
    return G, index