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

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

Python

>>> from sage.all import *
>>> ModularForms(Integer(11),Integer(2)).base_ring()
Rational Field
>>> ModularForms(Integer(1),Integer(12)).base_ring()
Rational Field
>>> CuspForms(Gamma1(Integer(13)),Integer(3)).base_ring()
Rational Field

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

Sage

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

Python

>>> from sage.all import *
>>> ModularForms(Integer(11),Integer(2),base_ring=GF(Integer(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

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

Python

>>> from sage.all import *
>>> ModularForms(DirichletGroup(Integer(13)).gen(0),Integer(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

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

Python

>>> from sage.all import *
>>> ModularForms(DirichletGroup(Integer(13)).gen(0)**Integer(6),Integer(3)).base_ring()
Rational Field

An example in characteristic \(7\):

Sage

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

Python

>>> from sage.all import *
>>> ModularForms(Integer(13),Integer(3),base_ring=GF(Integer(7))).base_ring()
Finite Field of size 7

AUTHORS:

William Stein (2007): first version

class sage.modular.modform.space.ModularFormsSpace(group, weight, character, base_ring, category=None)[source]#

Bases: HeckeModule_generic

A generic space of modular forms.

Element[source]#: alias of ModularFormElement

basis()[source]#

Return a basis for self.

EXAMPLES:

Sage

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)
]

Python

>>> from sage.all import *
>>> MM = ModularForms(Integer(11),Integer(2))
>>> 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()[source]#

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

sage: ModularForms(Gamma0(11),2).character()
Dirichlet character modulo 11 of conductor 1 mapping 2 |--> 1

Python

>>> from sage.all import *
>>> ModularForms(Gamma0(Integer(11)),Integer(2)).character()
Dirichlet character modulo 11 of conductor 1 mapping 2 |--> 1

Spaces of forms with nontrivial character:

Sage

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

Python

>>> from sage.all import *
>>> ModularForms(DirichletGroup(Integer(20)).gen(0),Integer(3)).character()
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1

>>> M = ModularForms(DirichletGroup(Integer(11)).gen(0), Integer(3))
>>> M.character()
Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta10
>>> s = M.cuspidal_submodule()
>>> s.character()
Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta10
>>> CuspForms(DirichletGroup(Integer(11)).gen(0),Integer(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

sage: print(ModularForms(Gamma1(11),2).character())
None

Python

>>> from sage.all import *
>>> print(ModularForms(Gamma1(Integer(11)),Integer(2)).character())
None

If the level is one then the character is trivial.

Sage

sage: ModularForms(Gamma1(1),12).character()
Dirichlet character modulo 1 of conductor 1

Python

>>> from sage.all import *
>>> ModularForms(Gamma1(Integer(1)),Integer(12)).character()
Dirichlet character modulo 1 of conductor 1

cuspidal_submodule()[source]#

Return the cuspidal submodule of self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> N = ModularForms(Integer(6),Integer(4)) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
>>> N.eisenstein_subspace().dimension()
4

Sage

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

Python

>>> from sage.all import *
>>> 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

sage: N.cuspidal_submodule().dimension()
1

Python

>>> from sage.all import *
>>> N.cuspidal_submodule().dimension()
1

We check that a bug noticed on Issue #10450 is fixed:

Sage

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

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(6), Integer(10))
>>> W = M.span_of_basis(M.basis()[Integer(0):Integer(2)])
>>> 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()[source]#

Synonym for cuspidal_submodule.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> N = ModularForms(Integer(6),Integer(4)) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
>>> N.eisenstein_subspace().dimension()
4

Sage

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

Python

>>> from sage.all import *
>>> 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

sage: N.cuspidal_submodule().dimension()
1

Python

>>> from sage.all import *
>>> N.cuspidal_submodule().dimension()
1

decomposition()[source]#

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

sage: M = ModularForms(11,2); M.decomposition()
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(11),Integer(2)); M.decomposition()
Traceback (most recent call last):
...
NotImplementedError

echelon_basis()[source]#

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

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)
]

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma0(Integer(11)),Integer(4))
>>> 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)
]
>>> 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

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)
]

Python

>>> from sage.all import *
>>> M = ModularForms(SL2Z, Integer(12))
>>> 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

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)
]

Python

>>> from sage.all import *
>>> M = CuspForms(Gamma0(Integer(17)),Integer(4), prec=Integer(10))
>>> 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()[source]#

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

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)
]

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(11))
>>> 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)
]
>>> 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

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)
]

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma1(Integer(6)),Integer(4))
>>> 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)
]
>>> 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

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)
]

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(11),Integer(4))
>>> 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)
]
>>> F = M.span_of_basis([M.gen(0) + Integer(1)/Integer(3)*M.gen(1), M.gen(2) + M.gen(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)
]
>>> 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()[source]#

Compute the Eisenstein series associated to this space.

Note

This function should be overridden by all derived classes.

EXAMPLES:

Sage

sage: # needs sage.rings.number_field
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

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> M = sage.modular.modform.space.ModularFormsSpace(Gamma0(Integer(11)), Integer(2), DirichletGroup(Integer(1))[Integer(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()[source]#

Return the Eisenstein submodule for this space of modular forms.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(11),Integer(2))
>>> 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 Issue #10450 is fixed:

Sage

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

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(6), Integer(10))
>>> W = M.span_of_basis(M.basis()[Integer(0):Integer(2)])
>>> 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()[source]#

Synonym for eisenstein_submodule.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(11),Integer(2))
>>> 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()[source]#

Return the underlying module of self.

EXAMPLES:

Sage

sage: N = ModularForms(6,4)
sage: N.dimension()
5

Python

>>> from sage.all import *
>>> N = ModularForms(Integer(6),Integer(4))
>>> N.dimension()
5

Sage

sage: N.embedded_submodule()
Vector space of dimension 5 over Rational Field

Python

>>> from sage.all import *
>>> N.embedded_submodule()
Vector space of dimension 5 over Rational Field

find_in_space(f, forms=None, prec=None, indep=True)[source]#

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

sage: M = ModularForms(11,2)
sage: N = ModularForms(10,2)
sage: M.find_in_space( M.basis()[0] )
[1, 0]

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(11),Integer(2))
>>> N = ModularForms(Integer(10),Integer(2))
>>> M.find_in_space( M.basis()[Integer(0)] )
[1, 0]

Sage

sage: M.find_in_space( N.basis()[0], forms=N.basis() )
[1, 0, 0]

Python

>>> from sage.all import *
>>> M.find_in_space( N.basis()[Integer(0)], forms=N.basis() )
[1, 0, 0]

Sage

sage: M.find_in_space( N.basis()[0] )
Traceback (most recent call last):
...
ArithmeticError: vector is not in free module

Python

>>> from sage.all import *
>>> M.find_in_space( N.basis()[Integer(0)] )
Traceback (most recent call last):
...
ArithmeticError: vector is not in free module

gen(n)[source]#

Return the nth generator of self.

EXAMPLES:

Sage

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)
]

Python

>>> from sage.all import *
>>> N = ModularForms(Integer(6),Integer(4))
>>> 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

sage: N.gen(0)
q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> N.gen(Integer(0))
q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6)

Sage

sage: N.gen(4)
q^3 + O(q^6)

Python

>>> from sage.all import *
>>> N.gen(Integer(4))
q^3 + O(q^6)

Sage

sage: N.gen(5)
Traceback (most recent call last):
...
ValueError: Generator 5 not defined

Python

>>> from sage.all import *
>>> N.gen(Integer(5))
Traceback (most recent call last):
...
ValueError: Generator 5 not defined

gens()[source]#

Return a complete set of generators for self.

EXAMPLES:

Sage

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)
]

Python

>>> from sage.all import *
>>> N = ModularForms(Integer(6),Integer(4))
>>> 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()[source]#

Return the congruence subgroup associated to this space of modular forms.

EXAMPLES:

Sage

sage: ModularForms(Gamma0(12),4).group()
Congruence Subgroup Gamma0(12)

Python

>>> from sage.all import *
>>> ModularForms(Gamma0(Integer(12)),Integer(4)).group()
Congruence Subgroup Gamma0(12)

Sage

sage: CuspForms(Gamma1(113),2).group()
Congruence Subgroup Gamma1(113)

Python

>>> from sage.all import *
>>> CuspForms(Gamma1(Integer(113)),Integer(2)).group()
Congruence Subgroup Gamma1(113)

Note that \(\Gamma_1(1)\) and \(\Gamma_0(1)\) are replaced by \(\SL_2(\ZZ)\).

Sage

sage: CuspForms(Gamma1(1),12).group()
Modular Group SL(2,Z)
sage: CuspForms(SL2Z,12).group()
Modular Group SL(2,Z)

Python

>>> from sage.all import *
>>> CuspForms(Gamma1(Integer(1)),Integer(12)).group()
Modular Group SL(2,Z)
>>> CuspForms(SL2Z,Integer(12)).group()
Modular Group SL(2,Z)

has_character()[source]#

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

sage: CuspForms(Gamma0(11),2).has_character()
True

Python

>>> from sage.all import *
>>> CuspForms(Gamma0(Integer(11)),Integer(2)).has_character()
True

A space for \(\Gamma_1(N)\) (for \(N\geq 2\)) never has a specific character.

Sage

sage: CuspForms(Gamma1(11),2).has_character()
False
sage: CuspForms(DirichletGroup(11).0,3).has_character()
True

Python

>>> from sage.all import *
>>> CuspForms(Gamma1(Integer(11)),Integer(2)).has_character()
False
>>> CuspForms(DirichletGroup(Integer(11)).gen(0),Integer(3)).has_character()
True

integral_basis()[source]#

Return an integral basis for this space of modular forms.

EXAMPLES:

In this example the integral and echelon bases are different.

Sage

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)
]

Python

>>> from sage.all import *
>>> m = ModularForms(Integer(97),Integer(2),prec=Integer(10))
>>> s = m.cuspidal_subspace()
>>> 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)
]
>>> 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

sage: m = CuspForms(64,2)
sage: m.integral_basis()
[
q + O(q^6),
q^2 + O(q^6),
q^5 + O(q^6)
]

Python

>>> from sage.all import *
>>> m = CuspForms(Integer(64),Integer(2))
>>> m.integral_basis()
[
q + O(q^6),
q^2 + O(q^6),
q^5 + O(q^6)
]

is_ambient()[source]#

Return True if this an ambient space of modular forms.

EXAMPLES:

Sage

sage: M = ModularForms(Gamma1(4),4)
sage: M.is_ambient()
True

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma1(Integer(4)),Integer(4))
>>> M.is_ambient()
True

Sage

sage: E = M.eisenstein_subspace()
sage: E.is_ambient()
False

Python

>>> from sage.all import *
>>> E = M.eisenstein_subspace()
>>> E.is_ambient()
False

is_cuspidal()[source]#

Return True if this space is cuspidal.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma0(Integer(11)), Integer(2)).new_submodule()
>>> M.is_cuspidal()
False
>>> M.cuspidal_submodule().is_cuspidal()
True

is_eisenstein()[source]#

Return True if this space is Eisenstein.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma0(Integer(11)), Integer(2)).new_submodule()
>>> M.is_eisenstein()
False
>>> M.eisenstein_submodule().is_eisenstein()
True

level()[source]#

Return the level of self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(47),Integer(3))
>>> M.level()
47

modular_symbols(sign=0)[source]#

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

sage: # needs sage.rings.number_field
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

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> M = sage.modular.modform.space.ModularFormsSpace(Gamma0(Integer(11)), Integer(2), DirichletGroup(Integer(1))[Integer(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)[source]#

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

sage: # needs sage.rings.number_field
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

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> M = sage.modular.modform.space.ModularFormsSpace(Gamma0(Integer(11)), Integer(2), DirichletGroup(Integer(1))[Integer(0)], base_ring=QQ); M.new_submodule()
Traceback (most recent call last):
...
NotImplementedError: computation of new submodule not yet implemented

new_subspace(p=None)[source]#

Synonym for new_submodule.

EXAMPLES:

Sage

sage: # needs sage.rings.number_field
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

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> M = sage.modular.modform.space.ModularFormsSpace(Gamma0(Integer(11)), Integer(2), DirichletGroup(Integer(1))[Integer(0)], base_ring=QQ); M.new_subspace()
Traceback (most recent call last):
...
NotImplementedError: computation of new submodule not yet implemented

newforms(names=None)[source]#

Return all newforms in the cuspidal subspace of self.

EXAMPLES:

Sage

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.

Python

>>> from sage.all import *
>>> CuspForms(Integer(18),Integer(4)).newforms()
[q + 2*q^2 + 4*q^4 - 6*q^5 + O(q^6)]
>>> CuspForms(Integer(32),Integer(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)]
>>> CuspForms(Integer(23)).newforms('b')
[q + b0*q^2 + (-2*b0 - 1)*q^3 + (-b0 - 1)*q^4 + 2*b0*q^5 + O(q^6)]
>>> CuspForms(Integer(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)[source]#

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

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)
]

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(1),Integer(12))
>>> S = M.cuspidal_subspace()
>>> S.prec()
6
>>> S.basis()
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]
>>> S.prec(Integer(8))
8
>>> 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)[source]#

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

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)
]

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(11),Integer(2))
>>> 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

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)
]

Python

>>> from sage.all import *
>>> 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)[source]#

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

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)
]

Python

>>> from sage.all import *
>>> S = ModularForms(Integer(11),Integer(2)).cuspidal_submodule()
>>> S.q_expansion_basis()
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
]
>>> S.q_expansion_basis(Integer(5))
[
q - 2*q^2 - q^3 + 2*q^4 + O(q^5)
]
>>> S = ModularForms(Integer(1),Integer(24)).cuspidal_submodule()
>>> S.q_expansion_basis(Integer(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

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)
]

Python

>>> from sage.all import *
>>> M = CuspForms(Integer(128), Integer(2), prec=Integer(3))
>>> 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)[source]#

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

sage: S = CuspForms(11,2)
sage: S.q_integral_basis(5)
[
q - 2*q^2 - q^3 + 2*q^4 + O(q^5)
]

Python

>>> from sage.all import *
>>> S = CuspForms(Integer(11),Integer(2))
>>> S.q_integral_basis(Integer(5))
[
q - 2*q^2 - q^3 + 2*q^4 + O(q^5)
]

set_precision(new_prec)[source]#

Set the default precision used for displaying \(q\)-expansions.

INPUT:

new_prec – positive integer

EXAMPLES:

Sage

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)
]

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma0(Integer(37)),Integer(2))
>>> M.set_precision(Integer(10))
>>> S = M.cuspidal_subspace()
>>> 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

sage: S.set_precision(0)
sage: S.basis()
[
O(q^0),
O(q^0)
]

Python

>>> from sage.all import *
>>> S.set_precision(Integer(0))
>>> S.basis()
[
O(q^0),
O(q^0)
]

The precision of subspaces is the same as the precision of the ambient space.

Sage

sage: S.set_precision(2)
sage: M.basis()
[
q + O(q^2),
O(q^2),
1 + 2/3*q + O(q^2)
]

Python

>>> from sage.all import *
>>> S.set_precision(Integer(2))
>>> M.basis()
[
q + O(q^2),
O(q^2),
1 + 2/3*q + O(q^2)
]

The precision must be nonnegative:

Sage

sage: S.set_precision(-1)
Traceback (most recent call last):
...
ValueError: n (=-1) must be >= 0

Python

>>> from sage.all import *
>>> S.set_precision(-Integer(1))
Traceback (most recent call last):
...
ValueError: n (=-1) must be >= 0

We do another example with nontrivial character.

Sage

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)

Python

>>> from sage.all import *
>>> M = ModularForms(DirichletGroup(Integer(13)).gen(0)**Integer(2))
>>> M.set_precision(Integer(10))
>>> M.cuspidal_subspace().gen(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)[source]#

Take a set B of forms, and return the subspace of self with B as a basis.

EXAMPLES:

Sage

sage: N = ModularForms(6,4) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field

Python

>>> from sage.all import *
>>> N = ModularForms(Integer(6),Integer(4)) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field

Sage

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

Python

>>> from sage.all import *
>>> N.span_of_basis([N.basis()[Integer(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

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

Python

>>> from sage.all import *
>>> N.span_of_basis([N.basis()[Integer(0)], N.basis()[Integer(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

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

Python

>>> from sage.all import *
>>> 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)[source]#

Take a set B of forms, and return the subspace of self with B as a basis.

EXAMPLES:

Sage

sage: N = ModularForms(6,4) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field

Python

>>> from sage.all import *
>>> N = ModularForms(Integer(6),Integer(4)) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field

Sage

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

Python

>>> from sage.all import *
>>> N.span_of_basis([N.basis()[Integer(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

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

Python

>>> from sage.all import *
>>> N.span_of_basis([N.basis()[Integer(0)], N.basis()[Integer(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

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

Python

>>> from sage.all import *
>>> 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)[source]#

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

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

Python

>>> from sage.all import *
>>> S37=CuspForms(Integer(37),Integer(2))
>>> S37.sturm_bound()
8
>>> M = ModularForms(Integer(11),Integer(2))
>>> M.sturm_bound()
3
>>> ModularForms(Gamma1(Integer(15)),Integer(2)).sturm_bound()
33

>>> CuspForms(Gamma1(Integer(144)), Integer(3)).sturm_bound()
3457
>>> CuspForms(DirichletGroup(Integer(144)).gen(1)**Integer(2), Integer(3)).sturm_bound()
73
>>> CuspForms(Gamma0(Integer(144)), Integer(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()[source]#

Return the weight of this space of modular forms.

EXAMPLES:

Sage

sage: M = ModularForms(Gamma1(13),11)
sage: M.weight()
11

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma1(Integer(13)),Integer(11))
>>> M.weight()
11

Sage

sage: M = ModularForms(Gamma0(997),100)
sage: M.weight()
100

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma0(Integer(997)),Integer(100))
>>> M.weight()
100

Sage

sage: M = ModularForms(Gamma0(97),4)
sage: M.weight()
4
sage: M.eisenstein_submodule().weight()
4

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma0(Integer(97)),Integer(4))
>>> M.weight()
4
>>> M.eisenstein_submodule().weight()
4

sage.modular.modform.space.contains_each(V, B)[source]#

Determine whether or not V contains every element of B. Used here for linear algebra, but works very generally.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> contains_each = sage.modular.modform.space.contains_each
>>> contains_each( range(Integer(20)), prime_range(Integer(20)) )
True
>>> contains_each( range(Integer(20)), range(Integer(30)) )
False

sage.modular.modform.space.is_ModularFormsSpace(x)[source]#

Return True if x is a `ModularFormsSpace`.

EXAMPLES:

Sage

sage: from sage.modular.modform.space import is_ModularFormsSpace
sage: is_ModularFormsSpace(ModularForms(11,2))
doctest:warning...
DeprecationWarning: The function is_ModularFormsSpace is deprecated; use 'isinstance(..., ModularFormsSpace)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
True
sage: is_ModularFormsSpace(CuspForms(11,2))
True
sage: is_ModularFormsSpace(3)
False

Python

>>> from sage.all import *
>>> from sage.modular.modform.space import is_ModularFormsSpace
>>> is_ModularFormsSpace(ModularForms(Integer(11),Integer(2)))
doctest:warning...
DeprecationWarning: The function is_ModularFormsSpace is deprecated; use 'isinstance(..., ModularFormsSpace)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
True
>>> is_ModularFormsSpace(CuspForms(Integer(11),Integer(2)))
True
>>> is_ModularFormsSpace(Integer(3))
False