space.py

r"""
Generic spaces of modular forms

EXAMPLES (computation of base ring): Return the base ring of this
space of modular forms.

EXAMPLES: For spaces of modular forms for `\Gamma_0(N)` or
`\Gamma_1(N)`, the default base ring is
`\QQ`::

    sage: ModularForms(11,2).base_ring()
    Rational Field
    sage: ModularForms(1,12).base_ring()
    Rational Field
    sage: CuspForms(Gamma1(13),3).base_ring()
    Rational Field            

The base ring can be explicitly specified in the constructor
function.

::

    sage: ModularForms(11,2,base_ring=GF(13)).base_ring()
    Finite Field of size 13

For modular forms with character the default base ring is the field
generated by the image of the character.

::

    sage: ModularForms(DirichletGroup(13).0,3).base_ring()
    Cyclotomic Field of order 12 and degree 4

For example, if the character is quadratic then the field is
`\QQ` (if the characteristic is `0`).

::

    sage: ModularForms(DirichletGroup(13).0^6,3).base_ring()
    Rational Field

An example in characteristic `7`::

    sage: ModularForms(13,3,base_ring=GF(7)).base_ring()
    Finite Field of size 7
"""

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

from sage.structure.all import Sequence

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

import sage.rings.all as rings

import defaults
import element
import hecke_operator_on_qexp
import submodule

import sage.modular.modform.constructor

from sage.matrix.constructor import zero_matrix
from sage.rings.arith import gcd

WARN=False

def is_ModularFormsSpace(x):
    r"""
    Return True if x is a ```ModularFormsSpace```.
    
    EXAMPLES::
    
        sage: from sage.modular.modform.space import is_ModularFormsSpace
        sage: is_ModularFormsSpace(ModularForms(11,2))
        True
        sage: is_ModularFormsSpace(CuspForms(11,2))
        True
        sage: is_ModularFormsSpace(3)
        False
    """
    return isinstance(x, ModularFormsSpace)

class ModularFormsSpace(hecke.HeckeModule_generic):
    """
    A generic space of modular forms.
    """
    def __init__(self, group, weight, character, base_ring):
        r"""
        Generic spaces of modular forms. For spaces of modular forms for
        `\Gamma_0(N)` or `\Gamma_1(N)`, the default base
        ring is `\QQ`.
        
        EXAMPLES::
        
            sage: ModularForms(11,2)
            Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
        
        ::
        
            sage: ModularForms(11,2,base_ring=GF(13))
            Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Finite Field of size 13
        
        ::
        
            sage: ModularForms(DirichletGroup(13).0,3)
            Modular Forms space of dimension 3, character [zeta12] and weight 3 over Cyclotomic Field of order 12 and degree 4
        
        ::
        
            sage: M = ModularForms(11,2)
            sage: M == loads(dumps(M))
            True
        """
        global WARN
        if WARN:
            print "Modular forms -- under development -- do not trust yet."
            WARN=False
        if not arithgroup.is_CongruenceSubgroup(group):
            raise TypeError, "group (=%s) must be a congruence subgroup"%group
        weight = int(weight)
        #if not isinstance(weight, int):
        #    raise TypeError, "weight must be an int"
        if not ((character is None) or isinstance(character, dirichlet.DirichletCharacter)):
            raise TypeError, "character must be a Dirichlet character"
        if not isinstance(base_ring, rings.Ring):
            raise TypeError, "base_ring must be a ring"
        self.__sturm_bound = None
        self.__weight, self.__group, self.__character = weight, group, character
        hecke.HeckeModule_generic.__init__(self, base_ring, group.level())

    def prec(self, new_prec=None):
        """
        Return or set the default precision used for displaying
        `q`-expansions of elements of this space.
        
        INPUT:
        
        
        -  ``new_prec`` - positive integer (default: None)
        
        
        OUTPUT: if new_prec is None, returns the current precision.
        
        EXAMPLES::
        
            sage: M = ModularForms(1,12)
            sage: S = M.cuspidal_subspace()
            sage: S.prec()
            6
            sage: S.basis()
            [
            q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
            ]
            sage: S.prec(8)
            8
            sage: S.basis()
            [
            q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + O(q^8)
            ]
        """
        return self.ambient().prec(new_prec)

    def set_precision(self, new_prec):
        """
        Set the default precision used for displaying
        `q`-expansions.
        
        INPUT:
        
        
        -  ``new_prec`` - positive integer
        
        
        EXAMPLES::
        
            sage: M = ModularForms(Gamma0(37),2)
            sage: M.set_precision(10)
            sage: S = M.cuspidal_subspace()
            sage: S.basis()
            [
            q + q^3 - 2*q^4 - q^7 - 2*q^9 + O(q^10),
            q^2 + 2*q^3 - 2*q^4 + q^5 - 3*q^6 - 4*q^9 + O(q^10)
            ]
        
        ::
        
            sage: S.set_precision(0)
            sage: S.basis()
            [
            O(q^0),
            O(q^0)
            ]
        
        The precision of subspaces is the same as the precision of the
        ambient space.
        
        ::
        
            sage: S.set_precision(2)
            sage: M.basis()
            [
            q + O(q^2),
            O(q^2),
            1 + 2/3*q + O(q^2)
            ]
        
        The precision must be nonnegative::
        
            sage: S.set_precision(-1)
            Traceback (most recent call last):
            ...
            ValueError: n (=-1) must be >= 0 
        
        We do another example with nontrivial character.
        
        ::
        
            sage: M = ModularForms(DirichletGroup(13).0^2)
            sage: M.set_precision(10)
            sage: M.cuspidal_subspace().0
            q + (-zeta6 - 1)*q^2 + (2*zeta6 - 2)*q^3 + zeta6*q^4 + (-2*zeta6 + 1)*q^5 + (-2*zeta6 + 4)*q^6 + (2*zeta6 - 1)*q^8 - zeta6*q^9 + O(q^10)
        """
        self.ambient().set_precision(new_prec)

    def change_ring(self, R):
        """
        Change the base ring of this space of modular forms. To be implemented in derived classes.
        
        EXAMPLES::
        
            sage: sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,QQ).change_ring(GF(7))
            Traceback (most recent call last):
            ...
            NotImplementedError: This function has not yet been implemented.
        """
        raise NotImplementedError, "This function has not yet been implemented."

    def weight(self):
        """
        Return the weight of this space of modular forms.
        
        EXAMPLES::
        
            sage: M = ModularForms(Gamma1(13),11)
            sage: M.weight()
            11
        
        ::
        
            sage: M = ModularForms(Gamma0(997),100)
            sage: M.weight()
            100
        
        ::
        
            sage: M = ModularForms(Gamma0(97),4)
            sage: M.weight()
            4
            sage: M.eisenstein_submodule().weight()
            4
        """
        return self.__weight

    def group(self):
        r"""
        Return the congruence subgroup associated to this space of modular
        forms.
        
        EXAMPLES::
        
            sage: ModularForms(Gamma0(12),4).group()
            Congruence Subgroup Gamma0(12)
        
        ::
        
            sage: CuspForms(Gamma1(113),2).group()
            Congruence Subgroup Gamma1(113)
        
        Note that `\Gamma_1(1)` and `\Gamma_0(1)` are replaced by
        `\mathrm{SL}_2(\ZZ)`.
        
        ::
        
            sage: CuspForms(Gamma1(1),12).group()
            Modular Group SL(2,Z)
            sage: CuspForms(SL2Z,12).group()
            Modular Group SL(2,Z)
        """
        return self.__group

    def character(self):
        """
        Return the Dirichlet character corresponding to this space of
        modular forms. Returns None if there is no specific character
        corresponding to this space, e.g., if this is a space of modular
        forms on `\Gamma_1(N)` with `N>1`.
        
        EXAMPLES: The trivial character::
        
            sage: ModularForms(Gamma0(11),2).character()
            Dirichlet character modulo 11 of conductor 1 mapping 2 |--> 1
        
        A space of forms with nontrivial character::
        
            sage: ModularForms(DirichletGroup(20).0,3).character()
            Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
        
        A space of forms with no particular character (hence None is
        returned)::
        
            sage: print ModularForms(Gamma1(11),2).character()
            None
        
        If the level is one then the character is trivial.
        
        ::
        
            sage: ModularForms(Gamma1(1),12).character()
            Dirichlet character modulo 1 of conductor 1 mapping 0 |--> 1
        """
        return self.__character

    def has_character(self):
        """
        Return True if this space of modular forms has a specific
        character.
        
        This is True exactly when the character() function does not return
        None.
        
        EXAMPLES: A space for `\Gamma_0(N)` has trivial character,
        hence has a character.
        
        ::
        
            sage: CuspForms(Gamma0(11),2).has_character()
            True
        
        A space for `\Gamma_1(N)` (for `N\geq 2`) never
        has a specific character.
        
        ::
        
            sage: CuspForms(Gamma1(11),2).has_character()
            False
            sage: CuspForms(DirichletGroup(11).0,3).has_character()
            True
        """
        return not self.character() is None
    
    def is_ambient(self):
        """
        Return True if this an ambient space of modular forms.
        
        EXAMPLES::
        
            sage: M = ModularForms(Gamma1(4),4)
            sage: M.is_ambient()
            True
        
        ::
        
            sage: E = M.eisenstein_subspace()
            sage: E.is_ambient()
            False
        """
        return False   # returning True is defined in the derived AmbientSpace class.

    def __normalize_prec(self, prec):
        """
        If prec=None, return self.prec(). Otherwise, make sure prec is a
        sensible choice of precision and return it.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4) 
            sage: N._ModularFormsSpace__normalize_prec(int(3))
            3
        
        ::
        
            sage: type(N._ModularFormsSpace__normalize_prec(int(3)))
            <type 'sage.rings.integer.Integer'>
        """
        if prec is None:
            prec = self.prec()
        else:
            prec = rings.Integer(prec)
        if prec < 0:
            raise ValueError, "prec (=%s) must be at least 0"%prec
        return prec

    def base_extend(self, base_ring):
        """
        Return the base extension of self to base_ring. This first checks
        whether there is a canonical coercion defined, and if so it calls the
        change_ring method.

        EXAMPLE::

            sage: N = ModularForms(6, 4)
            sage: N.base_extend(CyclotomicField(7))
            Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Cyclotomic Field of order 7 and degree 6

            sage: m = ModularForms(DirichletGroup(13).0^2,2); m
            Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2
            sage: m.base_extend(CyclotomicField(12))
            Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 12 and degree 4

            sage: chi = DirichletGroup(109, CyclotomicField(3)).0
            sage: S3 = CuspForms(chi, 2)
            sage: S9 = S3.base_extend(CyclotomicField(9))
            sage: S9
            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
            sage: S9.has_coerce_map_from(S3) # not implemented
            True
            sage: S9.base_extend(CyclotomicField(3))
            Traceback (most recent call last):
            ...
            ValueError: No coercion defined
        """
        if not base_ring.has_coerce_map_from(self.base_ring()):
            raise ValueError, "No coercion defined"
        else:
            return self.change_ring(base_ring)

    def echelon_form(self):
        r"""
        Return a space of modular forms isomorphic to self but with basis
        of `q`-expansions in reduced echelon form.
        
        This is useful, e.g., the default basis for spaces of modular forms
        is rarely in echelon form, but echelon form is useful for quickly
        recognizing whether a `q`-expansion is in the space.
        
        EXAMPLES: We first illustrate two ambient spaces and their echelon
        forms.
        
        ::
        
            sage: M = ModularForms(11)
            sage: M.basis()
            [
            q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
            1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + O(q^6)
            ]
            sage: M.echelon_form().basis()
            [
            1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + O(q^6),
            q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
            ]
        
        ::
        
            sage: M = ModularForms(Gamma1(6),4)
            sage: M.basis()
            [
            q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6),
            1 + O(q^6),
            q - 8*q^4 + 126*q^5 + O(q^6),
            q^2 + 9*q^4 + O(q^6),
            q^3 + O(q^6)
            ]
            sage: M.echelon_form().basis()
            [
            1 + O(q^6),
            q + 94*q^5 + O(q^6),
            q^2 + 36*q^5 + O(q^6),
            q^3 + O(q^6),
            q^4 - 4*q^5 + O(q^6)
            ]
        
        We create a space with a funny basis then compute the corresponding
        echelon form.
        
        ::
        
            sage: M = ModularForms(11,4)
            sage: M.basis()
            [
            q + 3*q^3 - 6*q^4 - 7*q^5 + O(q^6),
            q^2 - 4*q^3 + 2*q^4 + 8*q^5 + O(q^6),
            1 + O(q^6),
            q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
            ]
            sage: F = M.span_of_basis([M.0 + 1/3*M.1, M.2 + M.3]); F.basis()
            [
            q + 1/3*q^2 + 5/3*q^3 - 16/3*q^4 - 13/3*q^5 + O(q^6),
            1 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
            ]
            sage: E = F.echelon_form(); E.basis()
            [
            1 + 26/3*q^2 + 79/3*q^3 + 235/3*q^4 + 391/3*q^5 + O(q^6),
            q + 1/3*q^2 + 5/3*q^3 - 16/3*q^4 - 13/3*q^5 + O(q^6)
            ]
        """
        try:
            return self.__echelon_form
        except AttributeError:
            E = self.span_of_basis(self.echelon_basis())
            self.__echelon_form = E
            return E

    def echelon_basis(self):
        """
        Return a basis for self in reduced echelon form. This means that if
        we view the `q`-expansions of the basis as defining rows of
        a matrix (with infinitely many columns), then this matrix is in
        reduced echelon form.
        
        EXAMPLES::
        
            sage: M = ModularForms(Gamma0(11),4)
            sage: M.echelon_basis()
            [
            1 + O(q^6),
            q - 9*q^4 - 10*q^5 + O(q^6),
            q^2 + 6*q^4 + 12*q^5 + O(q^6),
            q^3 + q^4 + q^5 + O(q^6)
            ]
            sage: M.cuspidal_subspace().echelon_basis()
            [
            q + 3*q^3 - 6*q^4 - 7*q^5 + O(q^6),
            q^2 - 4*q^3 + 2*q^4 + 8*q^5 + O(q^6)
            ]
        
        ::
        
            sage: M = ModularForms(SL2Z, 12)
            sage: M.echelon_basis()
            [
            1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + O(q^6),
            q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
            ]
        
        ::
        
            sage: M = CuspForms(Gamma0(17),4, prec=10)
            sage: M.echelon_basis()
            [
            q + 2*q^5 - 8*q^7 - 8*q^8 + 7*q^9 + O(q^10),
            q^2 - 3/2*q^5 - 7/2*q^6 + 9/2*q^7 + q^8 - 4*q^9 + O(q^10),
            q^3 - 2*q^6 + q^7 - 4*q^8 - 2*q^9 + O(q^10),
            q^4 - 1/2*q^5 - 5/2*q^6 + 3/2*q^7 + 2*q^9 + O(q^10)
            ]
        """
        try:
            return self.__echelon_basis
        except AttributeError:
            F = self.free_module()
            W = self._q_expansion_module()
            pr = W.degree()
            B = self.q_echelon_basis(pr)
            E = [self(F.linear_combination_of_basis(W.coordinates(f.padded_list(pr)))) \
                              for f in B]
            E = Sequence(E, cr=True, immutable=True)
            self.__echelon_basis = E
            return E

    def integral_basis(self):
        """
        Return an integral basis for this space of modular forms.
        
        EXAMPLES: In this example the integral and echelon bases are
        different.
        
        ::
        
            sage: m = ModularForms(97,2,prec=10)
            sage: s = m.cuspidal_subspace()
            sage: s.integral_basis()
            [
            q + 2*q^7 + 4*q^8 - 2*q^9 + O(q^10),
            q^2 + q^4 + q^7 + 3*q^8 - 3*q^9 + O(q^10),
            q^3 + q^4 - 3*q^8 + q^9 + O(q^10),
            2*q^4 - 2*q^8 + O(q^10),
            q^5 - 2*q^8 + 2*q^9 + O(q^10),
            q^6 + 2*q^7 + 5*q^8 - 5*q^9 + O(q^10),
            3*q^7 + 6*q^8 - 4*q^9 + O(q^10)
            ]
            sage: s.echelon_basis()
            [
            q + 2/3*q^9 + O(q^10),
            q^2 + 2*q^8 - 5/3*q^9 + O(q^10),
            q^3 - 2*q^8 + q^9 + O(q^10),
            q^4 - q^8 + O(q^10),
            q^5 - 2*q^8 + 2*q^9 + O(q^10),
            q^6 + q^8 - 7/3*q^9 + O(q^10),
            q^7 + 2*q^8 - 4/3*q^9 + O(q^10)
            ]
        
        Here's another example where there is a big gap in the valuations::
        
            sage: m = CuspForms(64,2)
            sage: m.integral_basis()
            [
            q + O(q^6),
            q^2 + O(q^6),
            q^5 + O(q^6)
            ]        
        
        TESTS::
        
            sage: m = CuspForms(11*2^4,2, prec=13); m
            Cuspidal subspace of dimension 19 of Modular Forms space of dimension 30 for Congruence Subgroup Gamma0(176) of weight 2 over Rational Field
            sage: m.integral_basis()          # takes a long time (3 or 4 seconds)
            [
            q + O(q^13),
            q^2 + O(q^13),
            q^3 + O(q^13),
            q^4 + O(q^13),
            q^5 + O(q^13),
            q^6 + O(q^13),
            q^7 + O(q^13),
            q^8 + O(q^13),
            q^9 + O(q^13),
            q^10 + O(q^13),
            q^11 + O(q^13),
            q^12 + O(q^13),
            O(q^13),
            O(q^13),
            O(q^13),
            O(q^13),
            O(q^13),
            O(q^13),
            O(q^13)
            ]
        """
        try:
            return self.__integral_basis
        except AttributeError:
            W = self._q_expansion_module()
            pr = W.degree()
            B = self.q_integral_basis(pr)
            I = [self.linear_combination_of_basis(
                      W.coordinates(f.padded_list(pr))) for f in B]
            I = Sequence(I, cr=True, immutable=True)
            self.__integral_basis = I
            return I

    def _q_expansion_module(self):
        """
        Return module spanned by coefficients of q-expansions to sufficient
        precision to determine elements of this space.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2)
            sage: M._q_expansion_module()
            Vector space of degree 3 and dimension 2 over Rational Field
            User basis matrix:
            [   0    1   -2]
            [   1 12/5 36/5]
            sage: CuspForms(1,12)._q_expansion_module()
            Vector space of degree 2 and dimension 1 over Rational Field
            User basis matrix:
            [0 1]
        """
        try:
            return self.__q_expansion_module
        except AttributeError:
            pass

        prec = self.sturm_bound()
        C = self.q_expansion_basis(prec)
        V = self.base_ring()**prec
        W = V.span_of_basis([f.padded_list(prec) for f in C])
        self.__q_expansion_module = W
        return W
        
    def q_expansion_basis(self, prec=None):
        """
        Return a sequence of q-expansions for the basis of this space
        computed to the given input precision.
        
        INPUT:
        
        
        -  ``prec`` - integer (=0) or None
        
        
        If prec is None, the prec is computed to be *at least* large
        enough so that each q-expansion determines the form as an element
        of this space.
        
        .. note::

           In fact, the q-expansion basis is always computed to
           *at least* ``self.prec()``.
        
        EXAMPLES::
        
            sage: S = ModularForms(11,2).cuspidal_submodule()
            sage: S.q_expansion_basis()
            [
            q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
            ]            
            sage: S.q_expansion_basis(5)
            [
            q - 2*q^2 - q^3 + 2*q^4 + O(q^5)
            ]
            sage: S = ModularForms(1,24).cuspidal_submodule()
            sage: S.q_expansion_basis(8)
            [
            q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 - 982499328*q^6 - 147247240*q^7 + O(q^8),
            q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + 143820*q^6 - 985824*q^7 + O(q^8)
            ]
        """
        if prec is None:
            try: # don't care about precision -- just must be big enough to determine forms
                return self.__q_expansion_basis[1]
            except AttributeError:
                pass
            prec = -1  # big enough to determine forms
        else:
            prec = rings.Integer(self.__normalize_prec(prec))
            
        if prec == 0:
            z = self._q_expansion_ring()(0,prec)
            return Sequence([z]*int(self.dimension()), immutable=True, cr=True)
        elif prec != -1:
            try:
                current_prec, B = self.__q_expansion_basis
                if current_prec == prec:
                    return B
                elif current_prec > prec:
                    return Sequence([f.add_bigoh(prec) for f in B], immutable=True, cr=True)
            except AttributeError:
                pass

        d = self.dimension()
        current_prec = max(prec, self.prec(), int(1.2*d) + 3)         # +3 for luck.
        if prec == -1:
            prec = current_prec
        tries = 0
        while True:
            B = self._compute_q_expansion_basis(current_prec)
            if len(B) == d:
                break
            else:
                tries += 1
                current_prec += d
            if tries > 5:
                print "WARNING: possible bug in q_expansion_basis for modular forms space %s"%self
        B = Sequence(B, immutable=True, cr=True)
        self.__q_expansion_basis = (current_prec, B)
        if current_prec == prec:
            return B
        return Sequence([f.add_bigoh(prec) for f in B], immutable=True, cr=True)        

    def _compute_q_expansion_basis(self, prec):
        """
        EXAMPLES::
        
            sage: sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,QQ)._compute_q_expansion_basis(5)
            Traceback (most recent call last):
            ...
            NotImplementedError: this must be implemented in the derived class
        """
        raise NotImplementedError, "this must be implemented in the derived class"

    def q_echelon_basis(self, prec=None):
        r"""
        Return the echelon form of the basis of `q`-expansions of
        self up to precision prec.
        
        The `q`-expansions are power series (not actual modular
        forms). The number of `q`-expansions returned equals the
        dimension.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2)
            sage: M.q_expansion_basis()
            [
            q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
            1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + O(q^6)
            ]
        
        ::
        
            sage: M.q_echelon_basis()
            [
            1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + O(q^6),
            q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
            ]
        """
        prec = self.__normalize_prec(prec)
        if prec == 0:
            z = self._q_expansion_ring()(0,0)
            return Sequence([z]*int(self.dimension()), cr=True)
        try:
            current_prec, B = self.__q_echelon_basis
        except AttributeError:
            current_prec, B = -1, []
        if current_prec == prec:
            return B
        elif current_prec > prec:
            return Sequence([f.add_bigoh(prec) for f in B], cr=True)
        
        B = self.q_expansion_basis(prec)
        R = self.base_ring()
        A = R**prec
        gens = [f.padded_list(prec) for f in B]
        C = A.span(gens)
        
        T = self._q_expansion_ring()
        S = [T(f.list(), prec) for f in C.basis()]
        for _ in range(self.dimension() - len(S)):
            S.append(T(0,prec))
        S = Sequence(S, immutable=True, cr=True)
        self.__q_echelon_basis = (prec, S)
        return S
        
    def q_integral_basis(self, prec=None):
        r"""
        Return a `\ZZ`-reduced echelon basis of
        `q`-expansions for self.
        
        The `q`-expansions are power series with coefficients in
        `\ZZ`; they are *not* actual modular forms.
        
        The base ring of self must be `\QQ`. The number of
        `q`-expansions returned equals the dimension.
        
        EXAMPLES::
        
            sage: S = CuspForms(11,2)
            sage: S.q_integral_basis(5)
            [
            q - 2*q^2 - q^3 + 2*q^4 + O(q^5)
            ]
        """
        if not self.base_ring() == rings.QQ:
            raise TypeError, "the base ring must be Q"
        prec = self.__normalize_prec(prec)
        R = rings.PowerSeriesRing(rings.ZZ, name=defaults.DEFAULT_VARIABLE)
        if prec == 0:
            z = R(0,prec)
            return Sequence([z]*int(self.dimension()), cr=True)
        try:
            current_prec, B = self.__q_integral_basis
        except AttributeError:
            current_prec, B = -1, Sequence([], cr=True, immutable=True)
            
        if current_prec == prec:
            return B
        elif current_prec > prec:
            return Sequence([f.add_bigoh(prec) for f in B], cr=True)
        
        B = self.q_expansion_basis(prec)

        # It's over Q; we just need to intersect it with ZZ^n.
        A = rings.ZZ**prec
        zero = rings.ZZ(0)
        gens = [f.padded_list(prec) for f in B]        
        C = A.span(gens)
        D = C.saturation()
        S = [R(f.list(),prec) for f in D.basis()]
        for _ in range(self.dimension() - len(S)):
            S.append(R(0,prec))
        S = Sequence(S, immutable=True, cr=True)
        self.__q_integral_basis = (prec, S)
        return S

    def _q_expansion_ring(self):
        """
        Returns the parent for q-expansions of modular forms in self.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2)
            sage: M._q_expansion_ring()
            Power Series Ring in q over Rational Field
        """
        try:
            return self.__q_expansion_ring
        except AttributeError:
            R = rings.PowerSeriesRing(self.base_ring(), name=defaults.DEFAULT_VARIABLE)
            self.__q_expansion_ring = R
            return R

    def _q_expansion_zero(self):
        """
        Returns the q-expansion of the modular form 0.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2)
            sage: M._q_expansion_zero()
            0
            sage: M._q_expansion_zero() == M._q_expansion_ring()(0)
            True
        """
        try:
            return self.__q_expansion_zero
        except AttributeError:
            f = self._q_expansion_ring()(0)
            self.__q_expansion_zero = f
            return f

    def _q_expansion(self, element, prec):
        """
        Take an element of self (specified as a list, tuple, or vector),
        and return the corresponding q-expansion.
        
        EXAMPLES::
        
            sage: m = ModularForms(Gamma0(23),2); m
            Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(23) of weight 2 over Rational Field
            sage: m.basis()
            [
            q - q^3 - q^4 + O(q^6),
            q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6),
            1 + 12/11*q + 36/11*q^2 + 48/11*q^3 + 84/11*q^4 + 72/11*q^5 + O(q^6)
            ]
            sage: m._q_expansion([1,2,0], 5)
            q + 2*q^2 - 5*q^3 - 3*q^4 + O(q^5)
        """
        return self.ambient_module()._q_expansion(element, prec)

    def __add__(self, right):
        """
        If self and right live inside the same ambient module, return the
        sum of the two spaces (as modules).
        
        EXAMPLES::
        
            sage: N = CuspForms(44,2) ; ls = [N.submodule([N(u) for u in x.q_expansion_basis(20)]) for x in N.modular_symbols().decomposition()]; ls
            [Modular Forms subspace of dimension 1 of Modular Forms space of dimension 9 for Congruence Subgroup Gamma0(44) of weight 2 over Rational Field,
            Modular Forms subspace of dimension 3 of Modular Forms space of dimension 9 for Congruence Subgroup Gamma0(44) of weight 2 over Rational Field]

        ::
        
            sage: N1 = ls[0] ; N2 = ls[1]
            sage: N1 + N2 # indirect doctest
            Modular Forms subspace of dimension 4 of Modular Forms space of dimension 9 for Congruence Subgroup Gamma0(44) of weight 2 over Rational Field
        """
        from sage.modular.modform.submodule import ModularFormsSubmodule
        if self.ambient_module() != right.ambient_module():
            raise ArithmeticError, ("Sum of %s and %s not defined because " + \
                                    "they do not lie in a common ambient space.")%\
                                   (self, right)
        if self.is_ambient(): return self
        if right.is_ambient(): return right
        V = self.free_module() + right.free_module()
        return ModularFormsSubmodule(self.ambient_module(), V)
    
    def _has_natural_inclusion_map_to(self, right):
        """
        Return true if there is a natural inclusion map from modular forms
        in self to modular forms in right.
        
        INPUT:
        
        
        -  ``self, right`` - spaces of modular forms
        
        
        OUTPUT: True if self embeds in right, and False otherwise.
        
        TODO: Barring a few trivial cases, this only works in the case that
        right.is_ambient() returns True.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4) ; S = N.cuspidal_subspace()
        
        ::
        
            sage: N._has_natural_inclusion_map_to(S)
            Traceback (most recent call last):
            ...
            NotImplementedError
        
        ::
        
            sage: S._has_natural_inclusion_map_to(N)
            True
        
        ::
        
            sage: M = ModularForms(11,2)
            sage: N._has_natural_inclusion_map_to(M)
            False
        """
        if not right.group().is_subgroup(self.group()):
            return False
        if right.character() is None:
            # It's the full Gamma_1(N).
            return True
        if right.is_ambient():
            e = self.character()
            f = right.character()
            return f.parent()(e) == f
        raise NotImplementedError

    def has_coerce_map_from_impl(self, from_par):
        """
        Code to make ModularFormsSpace work well with coercion framework.
        
        EXAMPLES::
        
            sage: M = ModularForms(22,2)
            sage: M.has_coerce_map_from_impl(M.cuspidal_subspace())
            True
            sage: M.has_coerce_map_from(ModularForms(22,4))
            False
        """
        if isinstance(from_par, ModularFormsSpace):
            if from_par.ambient() == self:
                return True
            elif self.is_ambient() and self.group().is_subgroup(from_par.group()) and self.weight() == from_par.weight():
                return True
        
        return False

    def _coerce_impl(self, x):
        """
        Code to coerce an element into self.
        
        EXAMPLES::
        
            sage: M = ModularForms(22,2) ; S = CuspForms(22,2)
            sage: sum(S.basis()) 
            q + q^2 - q^3 - 4*q^4 + q^5 + O(q^6)
            sage: sum(S.basis() + M.basis())
            1 + 3*q + 3*q^2 + 2*q^3 - 7*q^4 + 8*q^5 + O(q^6)
            sage: M._coerce_impl(S.basis()[0])
            q - q^3 - 2*q^4 + q^5 + O(q^6)
        
        ::
        
            sage: M = ModularForms(Gamma0(22)) ; N = ModularForms(Gamma0(44))
            sage: M.basis()[0]
            q - q^3 - 2*q^4 + q^5 + O(q^6)
            sage: N(M.basis()[0])
            q - q^3 - 2*q^4 + q^5 + O(q^6)
        """
        if isinstance(x, element.ModularFormElement):
            if x.parent().ambient() == self:
                return self(x.element())
            elif self.group().is_subgroup(x.parent().group()):
                ## This is a coercion M_k(Gamma) --> M_k(Gamma'),
                ## where Gamma' is contained in Gamma.
                return self(x.q_expansion(self._q_expansion_module().degree()))

        raise TypeError, "no known coercion to modular form"
    
    def __call__(self, x, check=True):
        """
        Try to coerce x into self. If x is a vector of length
        self.dimension(), interpret it as a list of coefficients for
        self.basis() and return that linear combination. If x is a power
        series, it tries to determine whether or not x lives in self. If
        so, it returns x as an element of M, and throws an error if not.
        
        EXAMPLES::
        
            sage: M = ModularForms(13,4)
            sage: M.dimension()
            5
        
        ::
        
            sage: M([1,2,3,4,5])
            4 + 6*q + 47*q^2 + 143*q^3 + 358*q^4 + 630*q^5 + O(q^6)
        
        ::
        
            sage: M([1,3])
            Traceback (most recent call last):
            ...
            TypeError: entries must be a list of length 5
        
        ::
        
            sage: R = M._q_expansion_ring()
            sage: M(R([0,1,0,0,0,-2,-4,-2,-12]).add_bigoh(9))
            q - 2*q^5 + O(q^6)
        
        ::
        
            sage: M.set_precision(9)         
            sage: M(R([0,1,0,0,0,-2,-4,-2,-12]).add_bigoh(9))
            q - 2*q^5 - 4*q^6 - 2*q^7 - 12*q^8 + O(q^9)
        
        Note that one only needs coefficients up to self.sturm_bound() to
        determine the form::
        
            sage: M(R([0,1,0,0,0,-2,-4,-2,-12]).add_bigoh(8))
            q - 2*q^5 - 4*q^6 - 2*q^7 - 12*q^8 + O(q^9)
        
        ::
        
            sage: M(R([0,1,1,0,0,0,-4,-2,-12]).add_bigoh(9))
            Traceback (most recent call last):
            ...
            ValueError: q-expansion does not correspond to a form in self
        
        ::
        
            sage: S = CuspForms(1,12) ; R = PowerSeriesRing(QQ,'q') ; q = R.0
            sage: f = q+O(q^2) ; S(f)
            q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
            sage: f = q+2*q^2+O(q^3) ; S(f)
            Traceback (most recent call last):
            ...
            ValueError: q-expansion does not correspond to a form in self
            sage: f = q-24*q^2+O(q^3) ; S(f)
            q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
        """
        if isinstance(x, element.ModularFormElement):
            if x.parent() is self:
                return x
            
            if not check:
                from copy import copy
                f = copy(x)
                f._set_parent(self)
                return f

            try:
                if x.parent()._has_natural_inclusion_map_to(self.ambient()):
                    W = self._q_expansion_module()
                    return self(x.q_expansion(W.degree()))
            except NotImplementedError:
                pass
            raise TypeError, "unable to coerce x (= %s) into %s"%(x, self)
        
        elif rings.is_PowerSeries(x):
            W = self._q_expansion_module()
            if W.degree() <= x.prec():
                try:
                    x_potential = W.coordinates(x.padded_list(W.degree()))
                except ArithmeticError:
                    raise ValueError, "q-expansion does not correspond to a form in self" 
                x_potential = self.free_module().linear_combination_of_basis(x_potential)
                x_potential = element.ModularFormElement(self, x_potential)
                for i in range(int(W.degree()), x.prec()):
                    if x_potential[i] != x[i]:
                        raise ValueError, "q-expansion does not correspond to a form in self"
                return x_potential
            else:
                raise TypeError, "q-expansion needed to at least precision %s"%W.degree()
        return element.ModularFormElement(self, self.free_module()(x,check))
    
    def __cmp__(self, x):
        """
        Compare self and x.
        
        For spaces of modular forms, we order first by signature, then by
        dimension, and then by the ordering on the underlying free
        modules.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4) ; S = N.cuspidal_subspace()
            sage: S.__cmp__(N)
            -1
            sage: N.__cmp__(S)
            1
            sage: N.__cmp__(N)
            0
            sage: M = ModularForms(11,2)
            sage: N.__cmp__(M)
            -1
            sage: M.__cmp__(N)
            -1
        """
        from sage.modular.modform.constructor import canonical_parameters as params

        if self is x:
            return 0
        if not isinstance(x, ModularFormsSpace):
            return cmp( type(self), type(x) )
        
        left_ambient = self.ambient()
        right_ambient = x.ambient()
        if params(left_ambient.character(), left_ambient.level(),
                  left_ambient.weight(), left_ambient.base_ring()) != \
           params(right_ambient.character(), right_ambient.level(),
                  right_ambient.weight(), right_ambient.base_ring()):
            return -1
        if self.is_ambient() or x.is_ambient():
            return cmp(self.dimension(), x.dimension())
        else:
            return cmp(self.free_module(), x.free_module())

    def span_of_basis(self, B):
        """
        Take a set B of forms, and return the subspace of self with B as a
        basis.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4) ; N
            Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
        
        ::
        
            sage: N.span_of_basis([N.basis()[0]])
            Modular Forms subspace of dimension 1 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
        
        ::
        
            sage: N.span_of_basis([N.basis()[0], N.basis()[1]])
            Modular Forms subspace of dimension 2 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
        
        ::
        
            sage: N.span_of_basis( N.basis() )
            Modular Forms subspace of dimension 5 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
        """
        W = self._q_expansion_module()
        F = self.free_module()        
        prec = W.degree()
        C = [F.linear_combination_of_basis(W.coordinates(f.padded_list(prec))) for f in B]
        S = F.span_of_basis(C)
        return submodule.ModularFormsSubmoduleWithBasis(self.ambient(), S)
    
    span = span_of_basis

    def __submodule_from_subset_of_basis(self, x):
        """
        Return the submodule of self generated by the elements of x.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4)
            sage: N._ModularFormsSpace__submodule_from_subset_of_basis( [0,2] )
            Vector space of degree 5 and dimension 2 over Rational Field
            Basis matrix:
            [1 0 0 0 0]
            [0 0 1 0 0]
        """
        V = self.free_module()
        return V.submodule([V.gen(i) for i in x], check=False)
    
    def _compute_hecke_matrix_prime(self, p, prec=None):
        """
        Compute the matrix of the Hecke operator T_p acting on self.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2)
            sage: M._compute_hecke_matrix_prime(2)
            [-2  0]
            [ 0  3]            
        
        ::
        
            sage: M = ModularForms(11,2)
            sage: M2 = M.span([M.0 + M.1])
            sage: M2.hecke_matrix(2)
            Traceback (most recent call last):
            ...
            ArithmeticError: vector is not in free module
        """
        if prec is None:
            # Initial guess -- will increase if need be.
            # We add on a few dimensions, so we are likely to
            # detect non-invariant subspaces (if they accidentally occur).
            prec = p*self.dimension() + 8
        try:
            cur, _ = self.__q_expansion_basis
        except AttributeError:
            pass
        else:
            if prec < cur:
                prec = cur
        B = self.q_expansion_basis(prec)
        eps = self.character()
        if eps is None:
            raise NotImplementedError
        try:
            return hecke_operator_on_qexp.hecke_operator_on_basis(B, p,
                       self.weight(), eps, already_echelonized=False)
        except ValueError:
            # Double the precision.
            return self._compute_hecke_matrix_prime(p, prec = 2*prec+1)
        
    def _compute_hecke_matrix(self, n):
        """
        Compute the matrix of the Hecke operator T_n acting on self.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2)
            sage: M._compute_hecke_matrix(6)
            [ 2  0]
            [ 0 12]            
        
        ::
        
            sage: M = ModularForms(11,2)
            sage: M2 = M.span([M.0 + M.1])
            sage: M2.hecke_matrix(2)
            Traceback (most recent call last):
            ...
            ArithmeticError: vector is not in free module

        We check that #10450 is fixed::

            sage: M = CuspForms(Gamma1(22), 2).new_submodule()  # long time (3s on sage.math, 2011)
            sage: M.hecke_matrix(3)  # long time
            [ 0 -2  3  0]
            [ 0 -3  5 -1]
            [ 1 -1  0 -1]
            [ 0 -2  3 -1]
            sage: M.hecke_matrix(9)  # long time
            [ 3  3 -4 -4]
            [ 2  6 -9 -4]
            [ 0  3 -2 -1]
            [ 3  2 -7  0]
        """
        # For spaces with character, we calculate a basis of q-expansions and
        # use that. For Gamma1 and GammaH spaces, we would need to compute
        # diamond operators, which is quite slow; so we just compute on the
        # whole space and restrict.

        # TODO: If we know the subspace of the modular *symbols* space to which
        # this modular forms space corresponds, then that might give a quicker
        # way of doing this step.

        if hasattr(self, '_compute_q_expansion_basis') and self.character() is not None:
            return hecke.HeckeModule_generic._compute_hecke_matrix(self, n)

        else:
            # Try to avoid doing unnecessary computations where possible.
            if self.is_cuspidal():
                M = self.ambient().cuspidal_submodule().hecke_matrix(n).block_sum(zero_matrix(self.base_ring(), self.ambient().eisenstein_submodule().rank()))
            elif self.is_eisenstein():
                M = zero_matrix(self.base_ring(), self.ambient().cuspidal_submodule().rank()).block_sum(self.ambient().eisenstein_submodule().hecke_matrix(n))
            else:
                M = self.ambient().hecke_matrix(n)
            return M.restrict(self.free_module(), check=(gcd(n, self.level()) > 1))

    def basis(self):
        """
        Return a basis for self.
        
        EXAMPLES::
        
            sage: MM = ModularForms(11,2)
            sage: MM.basis()
            [
            q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
            1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + O(q^6)
            ]
        """
        try:
            return self.__basis
        except AttributeError:
            self.__basis = Sequence([element.ModularFormElement(self, x) for \
                                  x in self.free_module().basis()], immutable=True,
                                    cr = True)
        return self.__basis

    def gen(self, n):
        """
        Return the nth generator of self.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4)
            sage: N.basis()
            [
            q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6),
            1 + O(q^6),
            q - 8*q^4 + 126*q^5 + O(q^6),
            q^2 + 9*q^4 + O(q^6),
            q^3 + O(q^6)
            ]
        
        ::
        
            sage: N.gen(0)
            q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6)
        
        ::
        
            sage: N.gen(4)
            q^3 + O(q^6)
        
        ::
        
            sage: N.gen(5)
            Traceback (most recent call last):
            ...
            ValueError: Generator 5 not defined
        """
        try:
            return self.basis()[int(n)]
        except IndexError:
            raise ValueError, "Generator %s not defined"%n

    def gens(self):
        """
        Return a complete set of generators for self.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4)
            sage: N.gens()
            [
            q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6),
            1 + O(q^6),
            q - 8*q^4 + 126*q^5 + O(q^6),
            q^2 + 9*q^4 + O(q^6),
            q^3 + O(q^6)
            ]
        """
        return self.basis()
    
    def sturm_bound(self, M=None):
        r"""
        For a space M of modular forms, this function returns an integer B
        such that two modular forms in either self or M are equal if and
        only if their q-expansions are equal to precision B (note that this
        is 1+ the usual Sturm bound, since `O(q^\mathrm{prec})` has
        precision prec). If M is none, then M is set equal to self.
        
        EXAMPLES::
        
            sage: S37=CuspForms(37,2)
            sage: S37.sturm_bound()
            8
            sage: M = ModularForms(11,2)
            sage: M.sturm_bound()
            3
            sage: ModularForms(Gamma1(15),2).sturm_bound()
            33

            sage: CuspForms(Gamma1(144), 3).sturm_bound()
            3457
            sage: CuspForms(DirichletGroup(144).1^2, 3).sturm_bound()
            73
            sage: CuspForms(Gamma0(144), 3).sturm_bound()
            73
            
        REFERENCE:

        - [Sturm] J. Sturm, On the congruence of modular forms, Number theory 
          (New York, 1984-1985), Springer, Berlin, 1987, pp. 275-280.
        
        NOTE:
        
        Kevin Buzzard pointed out to me (William Stein) in Fall 2002 that
        the above bound is fine for Gamma1 with character, as one sees by
        taking a power of `f`. More precisely, if
        `f\cong 0\pmod{p}` for first `s` coefficients, then
        `f^r = 0 \pmod{p}` for first `s r` coefficients.
        Since the weight of `f^r` is
        `r \text{weight}(f)`, it follows that if
        `s \geq` the Sturm bound for `\Gamma_0` at
        weight(f), then `f^r` has valuation large enough to be
        forced to be `0` at `r\cdot` weight(f) by Sturm
        bound (which is valid if we choose `r` right). Thus
        `f \cong 0 \pmod{p}`. Conclusion: For `\Gamma_1`
        with fixed character, the Sturm bound is *exactly* the same as for
        `\Gamma_0`. A key point is that we are finding
        `\ZZ[\varepsilon]` generators for the Hecke algebra
        here, not `\ZZ`-generators. So if one wants
        generators for the Hecke algebra over `\ZZ`, this
        bound is wrong.
        
        This bound works over any base, even a finite field. There might be
        much better bounds over `\QQ`, or for comparing two
        eigenforms.
        """
        if M is not None:
            raise NotImplementedError
        if self.__sturm_bound is None:
            G = self.group()
            from sage.modular.arithgroup.all import is_Gamma1
            if is_Gamma1(G) and self.character() is not None:
                from sage.modular.arithgroup.all import Gamma0
                G = Gamma0(self.level())
            # the +1 below is because O(q^prec) has precision prec.
            self.__sturm_bound = G.sturm_bound(self.weight())+1
        return self.__sturm_bound             
    
    def character(self):
        """
        Return the Dirichlet character of this space.
        
        EXAMPLES::
        
            sage: M = ModularForms(DirichletGroup(11).0, 3)
            sage: M.character()
            Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta10
            sage: s = M.cuspidal_submodule()
            sage: s.character()
            Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta10
            sage: CuspForms(DirichletGroup(11).0,3).character()
            Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta10
        """
        return self.__character
    
    def cuspidal_submodule(self):
        """
        Return the cuspidal submodule of self.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4) ; N
            Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
            sage: N.eisenstein_subspace().dimension()
            4
        
        ::
        
            sage: N.cuspidal_submodule()
            Cuspidal subspace of dimension 1 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
        
        ::
        
            sage: N.cuspidal_submodule().dimension()
            1
        
        We check that a bug noticed on trac #10450 is fixed::
        
            sage: M = ModularForms(6, 10)
            sage: W = M.span_of_basis(M.basis()[0:2])
            sage: W.cuspidal_submodule()
            Modular Forms subspace of dimension 2 of Modular Forms space of dimension 11 for Congruence Subgroup Gamma0(6) of weight 10 over Rational Field
        """
        try:
            if self.__is_cuspidal == True:
                return self
            if self.__cuspidal_submodule != None:
                return self.__cuspidal_submodule
        except AttributeError:
            pass
        if self.is_ambient():
            # By definition the cuspidal submodule of the ambient space
            # is spanned by the first n standard basis vectors, where
            # n is the dimension of the cuspidal submodule.
            n = self.__ambient_cusp_dimension()
            W = self.__submodule_from_subset_of_basis(range(n))
            S = ModularForms(self, W)
            S.__is_cuspidal = True
            S.__is_eisenstein = (n==0)
            self.__cuspidal_submodule = S
            return S
        C = self.ambient_module().cuspidal_submodule()
        S = self.intersection(C)
        if S.dimension() < self.dimension():
            self.__is_cuspidal = False
            self.__cuspidal_submodule = S
        else:
            assert S.dimension() == self.dimension()
            self.__is_cuspidal = True
        S.__is_eisenstein = (S.dimension()==0)
        return S

    def cuspidal_subspace(self):
        """
        Synonym for cuspidal_submodule.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4) ; N
            Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
            sage: N.eisenstein_subspace().dimension()
            4
        
        ::
        
            sage: N.cuspidal_subspace()
            Cuspidal subspace of dimension 1 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
        
        ::
        
            sage: N.cuspidal_submodule().dimension()
            1
        """
        return self.cuspidal_submodule()

    def is_cuspidal(self):
        r"""
        Return True if this space is cuspidal.

        EXAMPLE::

            sage: M = ModularForms(Gamma0(11), 2).new_submodule()
            sage: M.is_cuspidal()
            False
            sage: M.cuspidal_submodule().is_cuspidal()
            True
        """
        return (self.cuspidal_submodule() == self)

    def is_eisenstein(self):
        r"""
        Return True if this space is Eisenstein.

        EXAMPLE::

            sage: M = ModularForms(Gamma0(11), 2).new_submodule()
            sage: M.is_eisenstein()
            False
            sage: M.eisenstein_submodule().is_eisenstein()
            True
        """
        return (self.eisenstein_submodule() == self)

    def new_submodule(self, p=None):
        """
         Return the new submodule of self. If p is specified, return the
         p-new submodule of self.
         
         .. note::

            This function should be overridden by all derived classes.
         
         EXAMPLES::
         
             sage: M = sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,base_ring=QQ) ; M.new_submodule()
             Traceback (most recent call last):
             ...
             NotImplementedError: computation of new submodule not yet implemented
         """
        raise NotImplementedError, "computation of new submodule not yet implemented"

    def new_subspace(self, p=None):
        """
        Synonym for new_submodule.
        
        EXAMPLES::
        
            sage: M = sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,base_ring=QQ) ; M.new_subspace()
            Traceback (most recent call last):
            ...
            NotImplementedError: computation of new submodule not yet implemented
        """
        return self.new_submodule(p)

    def eisenstein_series(self):
        """
        Compute the Eisenstein series associated to this space.
        
        .. note::

           This function should be overridden by all derived classes.
        
        EXAMPLES::
        
            sage: M = sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,base_ring=QQ) ; M.eisenstein_series()
            Traceback (most recent call last):
            ...
            NotImplementedError: computation of Eisenstein series in this space not yet implemented
        """
        raise NotImplementedError, "computation of Eisenstein series in this space not yet implemented"
        
    def decomposition(self):
        """
        This function returns a list of submodules `V(f_i,t)`
        corresponding to newforms `f_i` of some level dividing the
        level of self, such that the direct sum of the submodules equals
        self, if possible. The space `V(f_i,t)` is the image under
        `g(q)` maps to `g(q^t)` of the intersection with
        `R[[q]]` of the space spanned by the conjugates of
        `f_i`, where `R` is the base ring of self.
        
        TODO: Implement this function.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2); M.decomposition()
            Traceback (most recent call last):
            ...
            NotImplementedError
        """
        raise NotImplementedError        

    def newforms(self, names=None):
        """
        Return all newforms in the cuspidal subspace of self.
        
        EXAMPLES::
        
            sage: CuspForms(18,4).newforms()
            [q + 2*q^2 + 4*q^4 - 6*q^5 + O(q^6)]
            sage: CuspForms(32,4).newforms()
            [q - 8*q^3 - 10*q^5 + O(q^6), q + 22*q^5 + O(q^6), q + 8*q^3 - 10*q^5 + O(q^6)]
            sage: CuspForms(23).newforms('b') 
            [q + b0*q^2 + (-2*b0 - 1)*q^3 + (-b0 - 1)*q^4 + 2*b0*q^5 + O(q^6)] 
            sage: CuspForms(23).newforms() 
            Traceback (most recent call last): 
            ... 
            ValueError: Please specify a name to be used when generating names for generators of Hecke eigenvalue fields corresponding to the newforms.
        """
        M = self.modular_symbols(sign=1)
        factors = M.cuspidal_subspace().new_subspace().decomposition()
        large_dims = [ X.dimension() for X in factors if X.dimension() != 1 ]
        if len(large_dims) > 0 and names is None:
            raise ValueError, "Please specify a name to be used when generating names for generators of Hecke eigenvalue fields corresponding to the newforms."
        elif names is None:
            # In this case, we don't need a variable name, so insert 
            # something to get passed along below 
            names = 'a'
        return [ element.Newform(self, factors[i], names=(names+str(i)) )
                 for i in range(len(factors)) ]
        
    def eisenstein_submodule(self):
        """
        Return the Eisenstein submodule for this space of modular forms.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2)
            sage: M.eisenstein_submodule()
            Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

        We check that a bug noticed on trac #10450 is fixed::
        
            sage: M = ModularForms(6, 10)
            sage: W = M.span_of_basis(M.basis()[0:2])
            sage: W.eisenstein_submodule()
            Modular Forms subspace of dimension 0 of Modular Forms space of dimension 11 for Congruence Subgroup Gamma0(6) of weight 10 over Rational Field
        """
        try:
            if self.__is_eisenstein == True:
                return self
        except AttributeError:
            pass
        try:
            if self.__eisenstein_submodule != None:
                return self.__eisenstein_submodule
        except AttributeError:
            pass
        
        if self.is_ambient():
            # By definition the eisenstein submodule of the ambient space
            # is spanned by the n+1 through n+d standard basis vectors, where
            # n is the dimension of the cuspidal submodule and d
            # is the dimension of the eisenstein submodule (i.e., the
            # number of eisenstein series).
            n = self.__ambient_cusp_dimension()
            d = self.__ambient_eis_dimension()
            W = self.__submodule_from_subset_of_basis(range(n,n+d))
            E = ModularForms(self, W)
            E.__is_eisenstein = True
            E.__is_cuspidal = (d==0)
            self.__eisenstein_submodule = E
            return E
        A = self.ambient_module().eisenstein_submodule()
        E = self.intersection(A)
        if E.dimension() < self.dimension():
            self.__is_eisenstein = False
            self.__eisenstein_submodule = E
        else:
            assert E.dimension() == self.dimension()
            self.__is_eisenstein = True
        E.__is_cuspidal = (E.dimension()==0)
        return E
        
    def eisenstein_subspace(self):
        """
        Synonym for eisenstein_submodule.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2)
            sage: M.eisenstein_subspace()
            Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
        """
        return self.eisenstein_submodule()

    def embedded_submodule(self):
        """
        Return the underlying module of self.
        
        EXAMPLES::
        
            sage: N = ModularForms(6,4)
            sage: N.dimension()
            5
        
        ::
        
            sage: N.embedded_submodule()
            Vector space of dimension 5 over Rational Field
        """
        return self.free_module()

# intersect method commented out since it is a duplicate of the intersection method in sage.modular.hecke.submodule
# -- David Loeffler, 2009-04-30
#
#    def intersect(self, right):
#        """
#        If self and right live in the same ambient module, return the
#        intersection of self and right (as submodules).
#        
#        EXAMPLES::
#        
#            sage: N = ModularForms(6,4); S = N.cuspidal_subspace()
#        
#        ::
#        
#            sage: N.intersect(S)
#            Modular Forms subspace of dimension 1 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
#        
#        ::
#        
#            sage: S.intersect(N)
#            Modular Forms subspace of dimension 1 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
#        
#        ::
#        
#            sage: S.intersect(N.eisenstein_subspace())
#            Modular Forms subspace of dimension 0 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
#        """
#        from sage.modular.modform.all import ModularForms
#        if self.ambient_module() != right.ambient_module():
#            raise ArithmeticError, "Intersection of %s and %s not defined."%\
#                                   (self, right)
#        V = self.embedded_submodule().intersection(right.embedded_submodule())
##        return ModularForms(self.ambient_module(),V)
#        return self.span([ self(b) for b in V.basis() ])
    
##    def _key(self):
##        if self.is_ambient():
##            return self.__key
##        return self.__ambient
    
    def level(self):
        """
        Return the level of self.
        
        EXAMPLES::
        
            sage: M = ModularForms(47,3)
            sage: M.level()
            47
        """
        return self.group().level()
    
    def modular_symbols(self, sign=0):
        """
        Return the space of modular symbols corresponding to self with the
        given sign.
        
        .. note;:

           This function should be overridden by all derived classes.
        
        EXAMPLES::
        
            sage: M = sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,base_ring=QQ) ; M.modular_symbols()
            Traceback (most recent call last):
            ...
            NotImplementedError: computation of associated modular symbols space not yet implemented
        """
        raise NotImplementedError, "computation of associated modular symbols space not yet implemented"

    def find_in_space(self, f, forms=None, prec=None, indep=True):
        """
        INPUT:
        
        
        -  ``f`` - a modular form or power series
        
        -  ``forms`` - (default: None) a specific list of
           modular forms or q-expansions.
        
        -  ``prec`` - if forms are given, compute with them to
           the given precision
        
        -  ``indep`` - (default: True) whether the given list
           of forms are assumed to form a basis.
        
        
        OUTPUT: A list of numbers that give f as a linear combination of
        the basis for this space or of the given forms if
        independent=True.
        
        .. note::

           If the list of forms is given, they do *not* have to be in
           self.
        
        EXAMPLES::
        
            sage: M = ModularForms(11,2)
            sage: N = ModularForms(10,2)
            sage: M.find_in_space( M.basis()[0] )
            [1, 0]
        
        ::
        
            sage: M.find_in_space( N.basis()[0], forms=N.basis() )
            [1, 0, 0]
        
        ::
        
            sage: M.find_in_space( N.basis()[0] )
            Traceback (most recent call last):
            ...
            ArithmeticError: vector is not in free module
        """
        if forms is None or (forms == []):
            B = self._q_expansion_module()
            V = B.ambient_module()
            n = B.degree()
        else:
            if not isinstance(forms, (list, tuple)):
                raise TypeError, "forms must be a list or tuple"
            if prec is None:
                n = forms[0].parent().prec()
            else:
                n = prec
            V = self.base_ring()**n
            w = [V(g.padded_list(n)) for g in forms]
            if indep:
                B = V.span_of_basis(w)
            else:
                B = V.span(w)
        if rings.is_PowerSeries(f) and f.prec() < n:
            raise ValueError, "you need at least %s terms of precision"%n
        x = V(f.padded_list(n))
        return B.coordinates(x)


def contains_each(V, B):
    """
    Determine whether or not V contains every element of B. Used here
    for linear algebra, but works very generally.
    
    EXAMPLES::
    
        sage: contains_each = sage.modular.modform.space.contains_each
        sage: contains_each( range(20), prime_range(20) )
        True
        sage: contains_each( range(20), range(30) )
        False
    """
    for b in B:
        if not (b in V):
            return False
    return True