ambient_module.py

"""
Ambient Hecke modules
"""

#*****************************************************************************
#       Sage: System for Algebra and Geometry Experimentation
#
#       Copyright (C) 2005 William Stein <[email protected]>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#    This code is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#    General Public License for more details.
#
#  The full text of the GPL is available at:
#
#                  http://www.gnu.org/licenses/
#*****************************************************************************

import degenmap
import module
import submodule

import sage.modules.all

import sage.rings.all

import sage.misc.misc as misc

import sage.rings.arith as arith

import sage.matrix.matrix_space as matrix_space
from   sage.matrix.constructor import matrix

from sage.modular.arithgroup.all import Gamma0 # for Sturm bound

def is_AmbientHeckeModule(x):
    r"""
    Return True if x is of type AmbientHeckeModule.

    EXAMPLES::

        sage: from sage.modular.hecke.ambient_module import is_AmbientHeckeModule
        sage: is_AmbientHeckeModule(ModularSymbols(6))
        True
        sage: is_AmbientHeckeModule(ModularSymbols(6).cuspidal_subspace())
        False
        sage: is_AmbientHeckeModule(ModularForms(11))
        True
        sage: is_AmbientHeckeModule(BrandtModule(2, 3))
        True
    """
    return isinstance(x, AmbientHeckeModule)

class AmbientHeckeModule(module.HeckeModule_free_module):
    """
    An ambient Hecke module, i.e. a Hecke module that is isomorphic as a module
    over its base ring `R` to the standard free module `R^k` for some `k`. This
    is the base class for ambient spaces of modular forms and modular symbols,
    and for Brandt modules.
    """
    def __init__(self, base_ring, rank, level, weight):
        r"""
        Create an ambient Hecke module.

        EXAMPLES::

            sage: ModularSymbols(6) # indirect doctest
            Modular Symbols space of dimension 3 for Gamma_0(6) of weight 2 with sign 0 over Rational Field
            sage: sage.modular.hecke.ambient_module.AmbientHeckeModule(QQ, 3, 2, 4)
            Generic ambient Hecke module of rank 3, level 2 and weight 4 over Rational Field
        """
        rank = sage.rings.all.Integer(rank)
        if rank < 0:
            raise ValueError, "rank (=%s) must be nonnegative"%rank
        self.__rank = rank
        module.HeckeModule_free_module.__init__(self, base_ring, level, weight)

    def rank(self):
        """
        Return the rank of this ambient Hecke module.

        OUTPUT:

            Integer

        EXAMPLES::

            sage: M = sage.modular.hecke.ambient_module.AmbientHeckeModule(QQ, 3, 11, 2); M
            Generic ambient Hecke module of rank 3, level 11 and weight 2 over Rational Field
            sage: M.rank()
            3
        """
        return self.__rank

    def __add__(self, other):
        r"""
        Sum of self and other. As self is an ambient space, this will only make
        sense if other is a subspace of self, in which case the answer is self.

        EXAMPLES::

            sage: M = ModularSymbols(23)
            sage: M + M is M
            True
            sage: M + 3
            Traceback (most recent call last):
            ...
            TypeError: other (=3) must be a Hecke module.
        """
        if not isinstance(other, module.HeckeModule_free_module):
            raise TypeError, "other (=%s) must be a Hecke module."%other
        if other.ambient_hecke_module() == self:
            return self
        raise ArithmeticError, "Sum only defined for subspaces of a common ambient Hecke module."

    def _repr_(self):
        r"""
        String representation of self. Should be overridden by derived classes.

        EXAMPLE::

            sage: sage.modular.hecke.ambient_module.AmbientHeckeModule(QQ, 3, 2, 4)._repr_()
            'Generic ambient Hecke module of rank 3, level 2 and weight 4 over Rational Field'
        """
        return "Generic ambient Hecke module of rank %s, level %s and weight %s over %s"%(self.rank(), self.level(), self.weight(), self.base_ring())


    def _degeneracy_raising_matrix(self, codomain):
        """
        Matrix of the degeneracy map (with t = 1) from self to codomain, whose
        level should be a multiple of the level of self.

        EXAMPLE::

            sage: sage.modular.hecke.ambient_module.AmbientHeckeModule(QQ, 3, 2, 4)._degeneracy_raising_matrix(4)
            Traceback (most recent call last):
            ...
            NotImplementedError
        """
        raise NotImplementedError

    def _degeneracy_lowering_matrix(self, codomain, t):
        """
        Matrix of the degeneracy map of index t from self to codomain, whose level should be a divisor of the level of self.

        EXAMPLE::

            sage: sage.modular.hecke.ambient_module.AmbientHeckeModule(QQ, 3, 2, 4)._degeneracy_lowering_matrix(2, 2)
            Traceback (most recent call last):
            ...
            NotImplementedError
        """
        raise NotImplementedError

    def _hecke_image_of_ith_basis_element(self, n, i):
        """
        Return the image under the Hecke operator T_n of the i-th basis
        element.

        EXAMPLE::

            sage: sage.modular.hecke.ambient_module.AmbientHeckeModule(QQ, 3, 2, 4)._hecke_image_of_ith_basis_element(4, 2)
            Traceback (most recent call last):
            ...
            NotImplementedError

        """
        return self.hecke_operator(n)(self.gen(i))


    def _set_dual_free_module(self, V):
        r"""
        Store the embedded dual module of this module. Since this module is an
        ambient module, this is not necessary.

        EXAMPLE::

            sage: ModularForms(11, 2)._set_dual_free_module(None)
        """
        pass  # setting dual free module of ambient space is not necessary


    def ambient_hecke_module(self):
        r"""
        Return the ambient space that contains this ambient space. This is,
        of course, just this space again.

        EXAMPLE::

            sage: M = ModularForms(11, 4); M.ambient_hecke_module() is M
            True
        """
        return self

    def complement(self):
        """
        Return the largest Hecke-stable complement of this space.

        EXAMPLES::

            sage: M=ModularSymbols(11,2,1)
            sage: M
            Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
            sage: M.complement()
            Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
            sage: C=M.cuspidal_subspace()
            sage: C
            Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
            sage: C.complement()
            Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
        """
        return self.zero_submodule()

    def decomposition_matrix(self):
        r"""
        Returns the matrix whose columns form a basis for the canonical
        sorted decomposition of self coming from the Hecke operators.

        If the simple factors are `D_0, \ldots, D_n`, then the
        first few columns are an echelonized basis for `D_0`, the
        next an echelonized basis for `D_1`, the next for
        `D_2`, etc.

        EXAMPLE::

            sage: S = ModularSymbols(37, 2)
            sage: S.decomposition_matrix()
            [   1    0    0    0 -1/3]
            [   0    1   -1    0  1/2]
            [   0    0    0    1 -1/2]
            [   0    1    1    1    0]
            [   0    0    0    0    1]
        """
        try:
            return self.__decomposition_matrix_cache
        except AttributeError:
            rows = []
            for A in self.decomposition():
                for x in A.basis():
                    rows.append(x.list())
            A = matrix_space.MatrixSpace(self.base_ring(),self.rank())(rows)
            self.__decomposition_matrix_cache = A
            return self.__decomposition_matrix_cache

    def decomposition_matrix_inverse(self):
        """
        Returns the inverse of the matrix returned by
        decomposition_matrix().

        EXAMPLE::

            sage: S = ModularSymbols(37, 2)
            sage: t = S.decomposition_matrix_inverse(); t
            [   1    0    0    0  1/3]
            [   0  1/2 -1/2  1/2 -1/2]
            [   0 -1/2 -1/2  1/2    0]
            [   0    0    1    0  1/2]
            [   0    0    0    0    1]
            sage: t * S.decomposition_matrix() == 1
            True
        """
        try:
            return self.__decomposition_matrix_inverse_cache
        except AttributeError:
            self.__decomposition_matrix_inverse_cache = ~self.decomposition_matrix()
            return self.__decomposition_matrix_inverse_cache

    def degeneracy_map(self, codomain, t=1):
        """
        The `t`-th degeneracy map from self to the module ``codomain``.  The
        level of the codomain must be a divisor or multiple of level, and t
        must be a divisor of the quotient.

        INPUT:

        -  ``codomain`` - a Hecke module, which should be of the same type as
           self, or a positive integer (in which case Sage will use
           :meth:`~hecke_module_of_level` to find the "natural" module of the
           corresponding level).
        -  ``t`` - int, the parameter of the degeneracy map, i.e., the map is
           related to `f(q)` - `f(q^t)`.


        OUTPUT: A morphism from self to codomain.

        EXAMPLES::

            sage: M = ModularSymbols(11,sign=1)
            sage: d1 = M.degeneracy_map(33); d1
            Hecke module morphism degeneracy map corresponding to f(q) |--> f(q) defined by the matrix
            [ 1  0  0  0 -2 -1]
            [ 0  0 -2  2  0  0]
            Domain: Modular Symbols space of dimension 2 for Gamma_0(11) of weight ...
            Codomain: Modular Symbols space of dimension 6 for Gamma_0(33) of weight ...
            sage: M.degeneracy_map(33,3).matrix()
            [ 3  2  2  0 -2  1]
            [ 0  2  0 -2  0  0]
            sage: M = ModularSymbols(33,sign=1)
            sage: d2 = M.degeneracy_map(11); d2.matrix()
            [  1   0]
            [  0 1/2]
            [  0  -1]
            [  0   1]
            [ -1   0]
            [ -1   0]
            sage: (d2*d1).matrix()
            [4 0]
            [0 4]

        ::

            sage: M = ModularSymbols(3,12,sign=1)
            sage: M.degeneracy_map(1)
            Hecke module morphism degeneracy map corresponding to f(q) |--> f(q) defined by the matrix
            [1 0]
            [0 0]
            [0 1]
            [0 1]
            [0 1]
            Domain: Modular Symbols space of dimension 5 for Gamma_0(3) of weight ...
            Codomain: Modular Symbols space of dimension 2 for Gamma_0(1) of weight ...

        ::

            sage: S = M.cuspidal_submodule()
            sage: S.degeneracy_map(1)
            Hecke module morphism defined by the matrix
            [1 0]
            [0 0]
            [0 0]
            Domain: Modular Symbols subspace of dimension 3 of Modular Symbols space ...
            Codomain: Modular Symbols space of dimension 2 for Gamma_0(1) of weight ...

        ::

            sage: D = ModularSymbols(10,4).cuspidal_submodule().decomposition()
            sage: D
            [
            Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field,
            Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field
            ]
            sage: D[1].degeneracy_map(5)
            Hecke module morphism defined by the matrix
            [   0    0   -1    1]
            [   0  1/2  3/2   -2]
            [   0   -1    1    0]
            [   0 -3/4 -1/4    1]
            Domain: Modular Symbols subspace of dimension 4 of Modular Symbols space ...
            Codomain: Modular Symbols space of dimension 4 for Gamma_0(5) of weight ...

        We check for a subtle caching bug that came up in work on trac #10453::

            sage: loads(dumps(J0(33).decomposition()[0].modular_symbols()))
            Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field

        We check that certain absurd inputs are correctly caught::

            sage: chi = kronecker_character(7)
            sage: ModularSymbols(Gamma0(7), 4).degeneracy_map(ModularSymbols(chi, 4))
            Traceback (most recent call last):
            ...
            ValueError: The characters of the domain and codomain must match
        """
        if is_AmbientHeckeModule(codomain):
            M = codomain
            level = int(M.level())
        else:
            level = int(codomain)
            M = None

        t = int(t)

        err = False
        if self.level() % level == 0:
            quo = self.level() // level
            if quo % t != 0:
                err = True
        elif level % self.level() == 0:
            quo = level // self.level()
            if quo % t != 0:
                err = True
        else:
            err = True
        if err:
            raise ValueError, ("The level of self (=%s) must be a divisor or multiple of " + \
                               "level (=%s), and t (=%s) must be a divisor of the quotient.")%\
                               (self.level(), level, t)

        eps = self.character()
        if not (eps is None) and level % eps.conductor() != 0:
            raise ArithmeticError, "The conductor of the character of this space " + \
                  "(=%s) must be divisible by the level (=%s)."%\
                  (eps.conductor(), level)

        if M is None:
            M = self.hecke_module_of_level(level)

        if eps is not None and M.character() is not None:
            if eps.primitive_character() != M.character().primitive_character():
                raise ValueError("The characters of the domain and codomain must match")

        key = (M.group(), t)
        # bad idea to use (M, t) as the key, because using complicated objects
        # like modular forms spaces as dictionary keys causes weird behaviour;
        # on the other hand, (M.level(), t) isn't enough information.
        try:
            self._degeneracy_maps
        except AttributeError:
            self._degeneracy_maps = {}

        if self._degeneracy_maps.has_key(key):
            return self._degeneracy_maps[key]

        if M.rank() == 0:

            A = matrix_space.MatrixSpace(self.base_ring(), self.rank(),0)(0)

        elif self.level() % level == 0:  # lower the level

            A = self._degeneracy_lowering_matrix(M, t)

        elif level % self.level() == 0:  # raise the level

            A = self._degeneracy_raising_matrix(M, t)

        d = degenmap.DegeneracyMap(A, self, M, t)
        self._degeneracy_maps[key] = d
        return d

    def dual_free_module(self):
        r"""
        The free module dual to self, as a submodule of the dual
        module of the ambient space. As this space is ambient anyway,
        this just returns self.free_module().

        EXAMPLE::

            sage: M = ModularForms(2,8); M.dual_free_module()
            Vector space of dimension 3 over Rational Field
            sage: M.dual_free_module() is M.free_module()
            True
        """
        return self.free_module()

    def fcp(self, n, var='x'):
        """
        Returns the factorization of the characteristic polynomial of
        the Hecke operator `T_n` of index `n` acting on this space.

        INPUT:


        -  ``self`` - Hecke module invariant under the Hecke operator of index
           n.

        -  ``int n`` - a positive integer.

        -  ``var`` - variable of polynomial (default `x`)


        OUTPUT:

        -  ``list`` - list of the pairs (g,e), where g is an
           irreducible factor of the characteristic polynomial of T_n, and e
           is its multiplicity.

        EXAMPLES::

            sage: m = ModularSymbols(23, 2, sign=1)
            sage: m.fcp(2)
            (x - 3) * (x^2 + x - 1)
            sage: m.hecke_operator(2).charpoly('x').factor()
            (x - 3) * (x^2 + x - 1)
        """
        n = int(n)
        if n <= 0:
            raise ArithmeticError, "n (=%s) must be positive"%n
        return self.hecke_operator(n).fcp(var)

    def free_module(self):
        """
        Return the free module underlying this ambient Hecke module (the
        forgetful functor from Hecke modules to modules over the base ring)

        EXAMPLE::

            sage: ModularForms(59, 2).free_module()
            Vector space of dimension 6 over Rational Field
        """
        try:
            return self.__free_module
        except AttributeError:
            M = sage.modules.all.FreeModule(self.base_ring(), self.rank())
            self.__free_module = M
            return M

    def hecke_bound(self):
        r"""
        Return an integer B such that the Hecke operators `T_n`, for `n\leq B`,
        generate the full Hecke algebra as a module over the base ring. Note
        that we include the `n` with `n` not coprime to the level.

        At present this returns an unproven guess for non-cuspidal spaces which
        appears to be valid for `M_k(\Gamma_0(N))`, where k and N are the
        weight and level of self. (It is clearly valid for *cuspidal* spaces
        of any fixed character, as a consequence of the Sturm bound theorem.)
        It returns a hopelessly wrong answer for spaces of full level
        `\Gamma_1`.

        TODO: Get rid of this dreadful bit of code.

        EXAMPLE::

            sage: ModularSymbols(17, 4).hecke_bound()
            15
            sage: ModularSymbols(Gamma1(17), 4).hecke_bound() # wrong!
            15
        """
        try:
            if self.is_cuspidal():
                return Gamma0(self.level()).sturm_bound(self.weight())
        except AttributeError:
            pass
        misc.verbose("WARNING: ambient.py -- hecke_bound; returning unproven guess.")
        return Gamma0(self.level()).sturm_bound(self.weight()) + 2*Gamma0(self.level()).dimension_eis(self.weight()) + 5

    def hecke_module_of_level(self, level):
        r"""
        Return the Hecke module corresponding to self at the given level, which
        should be either a divisor or a multiple of the level of self. This
        raises NotImplementedError, and should be overridden in derived
        classes.

        EXAMPLE::

            sage: sage.modular.hecke.ambient_module.AmbientHeckeModule.hecke_module_of_level(ModularForms(2, 8),6)
            Traceback (most recent call last):
            ...
            NotImplementedError
        """
        raise NotImplementedError

    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(DirichletGroup(13).0, 3)
            sage: M.T(2)(M.0).element()
            (zeta12 + 4, 0, -1, 1)
            sage: M.hecke_images(0, [1,2])
            [         1          0          0          0]
            [zeta12 + 4          0         -1          1]
        """
        try:
            return self._hecke_images(i, v)
        except (AttributeError, NotImplementedError):
            pass
        # Use slow generic algorithm
        x = self.gen(i)
        X = [self.hecke_operator(n).apply_sparse(x).element() for n in v]
        return matrix(self.base_ring(), X)

    def intersection(self, other):
        """
        Returns the intersection of self and other, which must both lie in
        a common ambient space of modular symbols.

        EXAMPLES::

            sage: M = ModularSymbols(43, sign=1)
            sage: A = M[0] + M[1]
            sage: B = M[1] + M[2]
            sage: A.rank(), B.rank()
            (2, 3)
            sage: C = A.intersection(B); C.rank()  # TODO
            1
        """
        if not isinstance(other, module.HeckeModule_free_module):
            raise TypeError, "other (=%s) must be a Hecke module."%other
        if self.ambient_hecke_module() != other.ambient_hecke_module():
            raise ArithmeticError, "Intersection only defined for subspaces of a common ambient Hecke module."
        return other  # since self is ambient, so the intersection must equal other.

    def is_ambient(self):
        r"""
        Returns True if and only if self is an ambient Hecke module.

        .. warning::

           self can only be ambient by being of type
           AmbientHeckeModule.

           For example, decomposing a simple ambient space yields a
           single factor, and that factor is *not* considered an
           ambient space.

        EXAMPLES::

            sage: m = ModularSymbols(10)
            sage: m.is_ambient()
            True

        ::

            sage: a = m[0]  # the unique simple factor
            sage: a == m
            True
            sage: a.is_ambient()
            False
        """
        return True

    def is_full_hecke_module(self, compute=True):
        """
        Returns True if this space is invariant under the action of
        all Hecke operators, even those that divide the level. This is
        always true for ambient Hecke modules, so return True.

        EXAMPLE::

            sage: ModularSymbols(11, 4).is_full_hecke_module()
            True
        """
        return True

    def is_new(self, p=None):
        r"""
        Return True if this module is entirely new.

        EXAMPLE::

            sage: ModularSymbols(11, 4).is_new()
            False
            sage: ModularSymbols(1, 12).is_new()
            True
        """
        try:
            if self.__is_new.has_key(p):
                return self.__is_new[p]
        except AttributeError:
            pass
        AmbientHeckeModule.new_submodule(self,p)
        return self.__is_new[p]

    def is_old(self, p=None):
        r"""
        Return True if this module is entirely old.

        EXAMPLE::

            sage: ModularSymbols(22).is_old()
            True
            sage: ModularSymbols(3, 12).is_old()
            False
        """
        try:
            if self.__is_old.has_key(p):
                return self.__is_old[p]
        except AttributeError:
            pass
        self.old_submodule(p)
        return self.__is_old[p]

    def is_submodule(self, V):
        """
        Returns True if and only if self is a submodule of V. Since this is an
        ambient space, this returns True if and only if V is equal to self.

        EXAMPLE::

            sage: ModularSymbols(1, 4).is_submodule(ModularSymbols(11,4))
            False
            sage: ModularSymbols(11, 4).is_submodule(ModularSymbols(11,4))
            True
        """
        if not isinstance(V, module.HeckeModule_free_module):
            raise TypeError, "V must be a Hecke module"
        if not V.is_ambient():
            return False
        return V.ambient_hecke_module() == self

    def linear_combination_of_basis(self, v):
        r"""
        Given a list or vector of length equal to the dimension of self,
        construct the appropriate linear combination of the basis vectors of
        self.

        EXAMPLE::

            sage: ModularForms(3, 12).linear_combination_of_basis([1,0,0,0,1])
            2*q + 2049*q^2 + 177147*q^3 + 4196177*q^4 + 48830556*q^5 + O(q^6)

        """
        return self(v)

    def new_submodule(self, p=None):
        """
        Returns the new or p-new submodule of self.

        INPUT:

        -  ``p`` - (default: None); if not None, return only
           the p-new submodule.

        OUTPUT: the new or p-new submodule of self, i.e. the intersection of
        the kernel of the degeneracy lowering maps to level `N/p` (for the
        given prime `p`, or for all prime divisors of `N` if `p` is not given).

        If self is cuspidal this is a Hecke-invariant complement of the
        corresponding old submodule, but this may break down on Eisenstein
        subspaces (see the amusing example in William Stein's book of a form
        which is new and old at the same time).

        EXAMPLES::

            sage: m = ModularSymbols(33); m.rank()
            9
            sage: m.new_submodule().rank()
            3
            sage: m.new_submodule(3).rank()
            4
            sage: m.new_submodule(11).rank()
            8
        """
        try:
            if self.__is_new[p]:
                return self
        except AttributeError:
            self.__is_new = {}
        except KeyError:
            pass

        if self.rank() == 0:
            self.__is_new[p] = True
            return self
        try:
            return self.__new_submodule[p]
        except AttributeError:
            self.__new_submodule = {}
        except KeyError:
            pass

        # Construct the degeneracy map d.
        N = self.level()
        d = None
        eps = self.character()
        if eps == None:
            f = 1
        else:
            f = eps.conductor()
        if p == None:
            D = arith.prime_divisors(N)
        else:
            if N % p != 0:
                raise ValueError, "p must divide the level."
            D = [p]
        for q in D:
            # Here we are only using degeneracy *lowering* maps, so it is fine
            # to be careless and pass an integer for the level. One needs to be
            # a bit more careful with degeneracy *raising* maps for the Gamma1
            # and GammaH cases.
            if ((N//q) % f) == 0:
                NN = N//q
                d1 = self.degeneracy_map(NN,1).matrix()
                if d is None:
                    d = d1
                else:
                    d = d.augment(d1)
                d = d.augment(self.degeneracy_map(NN,q).matrix())
            #end if
        #end for
        if d is None or d == 0:
            self.__is_new[p] = True
            return self
        else:
            self.__is_new[p] = False
        ns = self.submodule(d.kernel(), check=False)
        ns.__is_new = {p:True}
        ns._is_full_hecke_module = True
        self.__new_submodule[p] = ns
        return ns

    def nonembedded_free_module(self):
        r"""
        Return the free module corresponding to self as an abstract free module
        (rather than as a submodule of an ambient free module). As this module
        is ambient anyway, this just returns ``self.free_module()``.

        EXAMPLES::

            sage: M = ModularSymbols(11, 2)
            sage: M.nonembedded_free_module() is M.free_module()
            True
        """
        return self.free_module()

    def old_submodule(self, p=None):
        """
        Returns the old or p-old submodule of self, i.e. the sum of the images
        of the degeneracy maps from level `N/p` (for the given prime `p`, or
        for all primes `p` dividing `N` if `p` is not given).

        INPUT:

        - ``p`` - (default: None); if not None, return only the p-old
          submodule.

        OUTPUT: the old or p-old submodule of self

        EXAMPLES::

            sage: m = ModularSymbols(33); m.rank()
            9
            sage: m.old_submodule().rank()
            7
            sage: m.old_submodule(3).rank()
            6
            sage: m.new_submodule(11).rank()
            8

        ::

            sage: e = DirichletGroup(16)([-1, 1])
            sage: M = ModularSymbols(e, 3, sign=1); M
            Modular Symbols space of dimension 4 and level 16, weight 3, character [-1, 1], sign 1, over Rational Field
            sage: M.old_submodule()
            Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 4 and level 16, weight 3, character [-1, 1], sign 1, over Rational Field

        Illustrate that trac 10664 is fixed::

            sage: ModularSymbols(DirichletGroup(42)[7], 6, sign=1).old_subspace(3)
            Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 40 and level 42, weight 6, character [-1, -1], sign 1, over Rational Field

        """
        try:
            if self.__is_old[p]:
                return self
        except AttributeError:
            self.__is_old = {}
        except KeyError:
            pass

        if self.rank() == 0:
            self.__is_old[p] = True
            return self
        try:
            return self.__old_submodule[p]
        except AttributeError:
            self.__old_submodule = {}
        except KeyError:
            pass

        # Construct the degeneracy map d.
        N = self.level()
        d = None

        eps = self.character()
        if eps is None:
            f = 1
        else:
            f = eps.conductor()

        if p is None:
            D = arith.prime_divisors(N)
        else:
            if N % p != 0:
                raise ValueError, "p must divide the level."
            D = [p]

        for q in D:
            NN = N//q
            if NN % f == 0:
                M = self.hecke_module_of_level(NN)

                # Here it is vital to pass self as an argument to
                # degeneracy_map, because M and the level N don't uniquely
                # determine self (e.g. the degeneracy map from level 1 to level
                # N could go to Gamma0(N), Gamma1(N) or anything in between)
                d1 = M.degeneracy_map(self, 1).matrix()

                if d is None:
                    d = d1
                else:
                    d = d.stack(d1)
                d = d.stack(M.degeneracy_map(self, q).matrix())
            #end if
        #end for
        if d is None:
            os = self.zero_submodule()
        else:
            os = self.submodule(d.image(), check=False)

        self.__is_old[p] = (os == self)

        os.__is_old = {p:True}
        os._is_full_hecke_module = True
        self.__old_submodule[p] = os
        return os

    def submodule(self, M, Mdual=None, check=True):
        """
        Return the Hecke submodule of self generated by M, which may be a
        submodule of the free module of self, or a list of elements of self.

        EXAMPLE::

            sage: M = ModularForms(37, 2)
            sage: A = M.submodule([M.newforms()[0].element(), M.newforms()[1].element()]); A
            Modular Forms subspace of dimension 2 of Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(37) of weight 2 over Rational Field
        """
        if check:
            if not sage.modules.all.is_FreeModule(M):
                V = self.free_module()
                if isinstance(M, (list,tuple)):
                    M = V.span([V(x.element()) for x in M])
                else:
                    M = V.span(M)
            if not M.is_submodule(self.free_module()):
                raise TypeError, "M must be a submodule of the free module associated to this module."
            if M == self.free_module():
                return self
        return self._submodule_class()(self, M, Mdual, check=check)

    def _submodule_class(self):
        r"""
        The class of submodules of this module. This is a separate method so it
        can be overridden in derived classes.

        EXAMPLE::

            sage: sage.modular.hecke.ambient_module.AmbientHeckeModule._submodule_class(ModularForms(1, 24))
            <class 'sage.modular.hecke.submodule.HeckeSubmodule'>
            sage: ModularForms(1, 24)._submodule_class()
            <class 'sage.modular.modform.submodule.ModularFormsSubmodule'>
        """
        return submodule.HeckeSubmodule

    def submodule_from_nonembedded_module(self, V, Vdual=None, check=True):
        """
        Create a submodule of this module, from a submodule of an ambient free
        module of the same rank as the rank of self.

        INPUT:

        -  ``V`` - submodule of ambient free module of the same rank as the
           rank of self.

        -  ``Vdual`` - used to pass in dual submodule (may be None)

        -  ``check`` - whether to check that submodule is Hecke equivariant

        OUTPUT: Hecke submodule of self

        EXAMPLE::

            sage: V = QQ^8
            sage: ModularForms(24, 2).submodule_from_nonembedded_module(V.submodule([0]))
            Modular Forms subspace of dimension 0 of Modular Forms space of dimension 8 for Congruence Subgroup Gamma0(24) of weight 2 over Rational Field
        """
        return self.submodule(V, Vdual, check=check)

    def submodule_generated_by_images(self, M):
        """
        Return the submodule of this ambient modular symbols space
        generated by the images under all degeneracy maps of M. The space M
        must have the same weight, sign, and group or character as this
        ambient space.

        EXAMPLES::

            sage: ModularSymbols(6, 12).submodule_generated_by_images(ModularSymbols(1,12))
            Modular Symbols subspace of dimension 12 of Modular Symbols space of dimension 22 for Gamma_0(6) of weight 12 with sign 0 over Rational Field
        """
        S = self.zero_submodule()
        if self.level() % M.level() == 0:
            D = arith.divisors(self.level() // M.level())
        elif M.level() % self.level() == 0:
            D = arith.divisors(M.level() // self.level())
        else:
            D = []
        for t in D:
            d = M.degeneracy_map(self, t)
            if d.codomain() != self:
                raise ArithmeticError, "incompatible spaces of modular symbols"
            S += d.image()

        if self.is_full_hecke_module(compute=False):
            S._is_full_hecke_module = True

        return S