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, category=None)

Bases: sage.modular.hecke.module.HeckeModule_generic

A generic space of modular forms.

Element

alias of sage.modular.modform.element.ModularFormElement

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)
]

character()

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

Spaces of forms with nontrivial character:

sage: ModularForms(DirichletGroup(20).0,3).character()
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1

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

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

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 ticket #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 ticket #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

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)
]

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.

EXAMPLES:

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.

EXAMPLES:

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.

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

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)
]

An example which used to be buggy:

sage: M = CuspForms(128, 2, prec=3)
sage: M.q_expansion_basis()
[
q - q^17 + O(q^22),
q^2 - 3*q^18 + O(q^22),
q^3 - q^11 + q^19 + O(q^22),
q^4 - 2*q^20 + O(q^22),
q^5 - 3*q^21 + O(q^22),
q^7 - q^15 + O(q^22),
q^9 - q^17 + O(q^22),
q^10 + O(q^22),
q^13 - q^21 + O(q^22)
]

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

REFERENCES:

• [Stu1987]

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