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# 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

class `sage.modular.modform.space.``ModularFormsSpace`(group, weight, character, base_ring)

  Bases: `sage.modular.hecke.module.HeckeModule_generic`

  A generic space of modular forms.

  `base_extend`(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

  `basis`()

    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)
        ]

  `change_ring`(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.

  `character`()

    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

  `cuspidal_submodule`()

    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

  `cuspidal_subspace`()

    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

  `decomposition`()

    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

  `echelon_basis`()

    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)
        ]

  `echelon_form`()

    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)
        ]

  `eisenstein_series`()

    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

  `eisenstein_submodule`()

    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

  `eisenstein_subspace`()

    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

  `embedded_submodule`()

    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

  `find_in_space`(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

  `gen`(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

  `gens`()

    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)
        ]

  `group`()

    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)

  `has_character`()

    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

  `has_coerce_map_from_impl`(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

  `integral_basis`()

    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)
        ]

  `is_ambient`()

    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

  `is_cuspidal`()

    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

  `is_eisenstein`()

    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

  `level`()

    Return the level of self.

    EXAMPLES:

        sage: M = ModularForms(47,3)
        sage: M.level()
        47

  `modular_symbols`(sign=0)

    Return the space of modular symbols corresponding to self with the given sign.

    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

  `new_submodule`(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

  `new_subspace`(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

  `newforms`(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.

  `prec`(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)
        ]

  `q_echelon_basis`(prec=None)

    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)
        ]

  `q_expansion_basis`(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)
        ]

  `q_integral_basis`(prec=None)

    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)
        ]

  `set_precision`(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)

  `span`(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

  `span_of_basis`(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

  `sturm_bound`(M=None)

    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.

  `weight`()

    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

`sage.modular.modform.space.``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

`sage.modular.modform.space.``is_ModularFormsSpace`(x)

  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

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