Hecke modules

class sage.modular.hecke.module.HeckeModule_free_module(base_ring, level, weight, category=None)[source]

Bases: HeckeModule_generic

A Hecke module modeled on a free module over a commutative ring.

T(n)[source]

Return the \(n\)-th Hecke operator \(T_n\).

This function is a synonym for hecke_operator().

EXAMPLES:

sage: M = ModularSymbols(11,2)
sage: M.T(3)
Hecke operator T_3 on Modular Symbols ...

>>> from sage.all import *
>>> M = ModularSymbols(Integer(11),Integer(2))
>>> M.T(Integer(3))
Hecke operator T_3 on Modular Symbols ...
ambient()[source]

Return the ambient module associated to this module.

Synonym for ambient_hecke_module().

EXAMPLES:

sage: CuspForms(1, 12).ambient()
Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field

>>> from sage.all import *
>>> CuspForms(Integer(1), Integer(12)).ambient()
Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
ambient_hecke_module()[source]

Return the ambient module associated to this module.

As this is an abstract base class, raise NotImplementedError.

EXAMPLES:

sage: sage.modular.hecke.module.HeckeModule_free_module(QQ, 10, 3).ambient_hecke_module()
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> sage.modular.hecke.module.HeckeModule_free_module(QQ, Integer(10), Integer(3)).ambient_hecke_module()
Traceback (most recent call last):
...
NotImplementedError
ambient_module()[source]

Return the ambient module associated to this module.

Synonym for ambient_hecke_module().

EXAMPLES:

sage: CuspForms(1, 12).ambient_module()
Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
sage: sage.modular.hecke.module.HeckeModule_free_module(QQ, 10, 3).ambient_module()
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> CuspForms(Integer(1), Integer(12)).ambient_module()
Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
>>> sage.modular.hecke.module.HeckeModule_free_module(QQ, Integer(10), Integer(3)).ambient_module()
Traceback (most recent call last):
...
NotImplementedError
atkin_lehner_operator(d=None)[source]

Return the Atkin-Lehner operator \(W_d\) on this space, if defined, where \(d\) is a divisor of the level \(N\) such that \(N/d\) and \(d\) are coprime. If \(d\) is not given, we take \(d = N\). If \(N/d\) is not coprime to \(d\), then we replace \(d\) with the unique integer having this property which has the same prime factors as \(d\).

Note

The operator \(W_d\) is given by the action of any matrix of the form

\[\begin{split}W_d = \begin{pmatrix} dx & y \\ Nz & dw \end{pmatrix}\end{split}\]

with \(\det W_d = d\) and such that \(x = 1 \bmod N/d\), \(y = 1 \bmod d\), as in [AL1978]. However, our definition of the weight \(k\) action differs from theirs by a power of the determinant, so our operator \(W_d\) is \(d^{k/2 - 1}\) times the operator of Atkin-Li. In particular, if \(k = 2\) our conventions are identical to Atkin and Li’s.

With Sage’s conventions, the operator \(W_d\) satisfies

\[W_d^2 = d^{k - 2} \langle x^{-1} \rangle\]

where \(x\) is congruent to \(d\) modulo \(N/d\) and to \(-1\) modulo \(d\). In particular, the operator is an involution in weight 2 and trivial character (but not in most other situations).

EXAMPLES:

sage: M = ModularSymbols(11)
sage: w = M.atkin_lehner_operator()
sage: w
Hecke module morphism Atkin-Lehner operator W_11 defined by the matrix
[-1  0  0]
[ 0 -1  0]
[ 0  0 -1]
Domain: Modular Symbols space of dimension 3 for Gamma_0(11) of weight ...
Codomain: Modular Symbols space of dimension 3 for Gamma_0(11) of weight ...
sage: M = ModularSymbols(Gamma1(13))
sage: w = M.atkin_lehner_operator()
sage: w.fcp('x')
(x - 1)^7 * (x + 1)^8

>>> from sage.all import *
>>> M = ModularSymbols(Integer(11))
>>> w = M.atkin_lehner_operator()
>>> w
Hecke module morphism Atkin-Lehner operator W_11 defined by the matrix
[-1  0  0]
[ 0 -1  0]
[ 0  0 -1]
Domain: Modular Symbols space of dimension 3 for Gamma_0(11) of weight ...
Codomain: Modular Symbols space of dimension 3 for Gamma_0(11) of weight ...
>>> M = ModularSymbols(Gamma1(Integer(13)))
>>> w = M.atkin_lehner_operator()
>>> w.fcp('x')
(x - 1)^7 * (x + 1)^8


sage: M = ModularSymbols(33)
sage: S = M.cuspidal_submodule()
sage: S.atkin_lehner_operator()
Hecke module morphism Atkin-Lehner operator W_33 defined by the matrix
[ 0 -1  0  1 -1  0]
[ 0 -1  0  0  0  0]
[ 0 -1  0  0 -1  1]
[ 1 -1  0  0 -1  0]
[ 0  0  0  0 -1  0]
[ 0 -1  1  0 -1  0]
Domain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...

>>> from sage.all import *
>>> M = ModularSymbols(Integer(33))
>>> S = M.cuspidal_submodule()
>>> S.atkin_lehner_operator()
Hecke module morphism Atkin-Lehner operator W_33 defined by the matrix
[ 0 -1  0  1 -1  0]
[ 0 -1  0  0  0  0]
[ 0 -1  0  0 -1  1]
[ 1 -1  0  0 -1  0]
[ 0  0  0  0 -1  0]
[ 0 -1  1  0 -1  0]
Domain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...


sage: S.atkin_lehner_operator(3)
Hecke module morphism Atkin-Lehner operator W_3 defined by the matrix
[ 0  1  0 -1  1  0]
[ 0  1  0  0  0  0]
[ 0  1  0  0  1 -1]
[-1  1  0  0  1  0]
[ 0  0  0  0  1  0]
[ 0  1 -1  0  1  0]
Domain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...

>>> from sage.all import *
>>> S.atkin_lehner_operator(Integer(3))
Hecke module morphism Atkin-Lehner operator W_3 defined by the matrix
[ 0  1  0 -1  1  0]
[ 0  1  0  0  0  0]
[ 0  1  0  0  1 -1]
[-1  1  0  0  1  0]
[ 0  0  0  0  1  0]
[ 0  1 -1  0  1  0]
Domain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 6 of Modular Symbols space ...


sage: N = M.new_submodule()
sage: N.atkin_lehner_operator()
Hecke module morphism Atkin-Lehner operator W_33 defined by the matrix
[  1 2/5 4/5]
[  0  -1   0]
[  0   0  -1]
Domain: Modular Symbols subspace of dimension 3 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 3 of Modular Symbols space ...

>>> from sage.all import *
>>> N = M.new_submodule()
>>> N.atkin_lehner_operator()
Hecke module morphism Atkin-Lehner operator W_33 defined by the matrix
[  1 2/5 4/5]
[  0  -1   0]
[  0   0  -1]
Domain: Modular Symbols subspace of dimension 3 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 3 of Modular Symbols space ...
basis()[source]

Return a basis for self.

EXAMPLES:

sage: m = ModularSymbols(43)
sage: m.basis()
((1,0), (1,31), (1,32), (1,38), (1,39), (1,40), (1,41))

>>> from sage.all import *
>>> m = ModularSymbols(Integer(43))
>>> m.basis()
((1,0), (1,31), (1,32), (1,38), (1,39), (1,40), (1,41))
basis_matrix()[source]

Return the matrix of the basis vectors of self (as vectors in some ambient module)

EXAMPLES:

sage: CuspForms(1, 12).basis_matrix()
[1 0]

>>> from sage.all import *
>>> CuspForms(Integer(1), Integer(12)).basis_matrix()
[1 0]
coordinate_vector(x)[source]

Write x as a vector with respect to the basis given by self.basis().

EXAMPLES:

sage: S = ModularSymbols(11,2).cuspidal_submodule()
sage: S.0
(1,8)
sage: S.basis()
((1,8), (1,9))
sage: S.coordinate_vector(S.0)
(1, 0)

>>> from sage.all import *
>>> S = ModularSymbols(Integer(11),Integer(2)).cuspidal_submodule()
>>> S.gen(0)
(1,8)
>>> S.basis()
((1,8), (1,9))
>>> S.coordinate_vector(S.gen(0))
(1, 0)
decomposition(bound=None, anemic=True, height_guess=1, sort_by_basis=False, proof=None)[source]

Return the maximal decomposition of this Hecke module under the action of Hecke operators of index coprime to the level.

This is the finest decomposition of self that we can obtain using factors obtained by taking kernels of Hecke operators.

Each factor in the decomposition is a Hecke submodule obtained as the kernel of \(f(T_n)^r\) acting on self, where n is coprime to the level and \(r=1\). If anemic is False, instead choose \(r\) so that \(f(X)^r\) exactly divides the characteristic polynomial.

INPUT:

  • anemic – boolean (default: True); if True, use only Hecke operators of index coprime to the level

  • bound – integer or None (default: None); if None, use all Hecke operators up to the Sturm bound, and hence obtain the same result as one would obtain by using every element of the Hecke ring. If a fixed integer, decompose using only Hecke operators \(T_p\), with \(p\) prime, up to bound.

  • sort_by_basis – boolean (default: False); if True the resulting decomposition will be sorted as if it was free modules, ignoring the Hecke module structure. This will save a lot of time.

OUTPUT: list of subspaces of self

EXAMPLES:

sage: ModularSymbols(17,2).decomposition()
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field
]
sage: ModularSymbols(Gamma1(10),4).decomposition()
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field
]
sage: ModularSymbols(GammaH(12, [11])).decomposition()
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 5 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field
]

>>> from sage.all import *
>>> ModularSymbols(Integer(17),Integer(2)).decomposition()
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field
]
>>> ModularSymbols(Gamma1(Integer(10)),Integer(4)).decomposition()
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 over Rational Field
]
>>> ModularSymbols(GammaH(Integer(12), [Integer(11)])).decomposition()
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 5 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 over Rational Field
]
degree()[source]

Return the degree of this Hecke module.

This is the rank of the ambient free module.

EXAMPLES:

sage: CuspForms(1, 12).degree()
2

>>> from sage.all import *
>>> CuspForms(Integer(1), Integer(12)).degree()
2
diamond_bracket_matrix(d)[source]

Return the matrix of the diamond bracket operator \(\langle d \rangle\) on self.

EXAMPLES:

sage: M = ModularSymbols(DirichletGroup(5).0, 3)
sage: M.diamond_bracket_matrix(3)
[-zeta4      0]
[     0 -zeta4]
sage: ModularSymbols(Gamma1(5), 3).diamond_bracket_matrix(3)
[ 0  1  0  0]
[-1  0  0  0]
[ 0  0  0  1]
[ 0  0 -1  0]

>>> from sage.all import *
>>> M = ModularSymbols(DirichletGroup(Integer(5)).gen(0), Integer(3))
>>> M.diamond_bracket_matrix(Integer(3))
[-zeta4      0]
[     0 -zeta4]
>>> ModularSymbols(Gamma1(Integer(5)), Integer(3)).diamond_bracket_matrix(Integer(3))
[ 0  1  0  0]
[-1  0  0  0]
[ 0  0  0  1]
[ 0  0 -1  0]
diamond_bracket_operator(d)[source]

Return the diamond bracket operator \(\langle d \rangle\) on self.

EXAMPLES:

sage: M = ModularSymbols(DirichletGroup(5).0, 3)
sage: M.diamond_bracket_operator(3)
Diamond bracket operator <3> on Modular Symbols space of dimension 2 and level 5, weight 3, character [zeta4], sign 0, over Cyclotomic Field of order 4 and degree 2

>>> from sage.all import *
>>> M = ModularSymbols(DirichletGroup(Integer(5)).gen(0), Integer(3))
>>> M.diamond_bracket_operator(Integer(3))
Diamond bracket operator <3> on Modular Symbols space of dimension 2 and level 5, weight 3, character [zeta4], sign 0, over Cyclotomic Field of order 4 and degree 2
dual_eigenvector(names='alpha', lift=True, nz=None)[source]

Return an eigenvector for the Hecke operators acting on the linear dual of this space.

This eigenvector will have entries in an extension of the base ring of degree equal to the dimension of this space.

Warning

The input space must be simple.

INPUT:

  • name – print name of generator for eigenvalue field

  • lift – boolean (default: True)

  • nz – if not None, then normalize vector so dot product with this basis vector of ambient space is 1

OUTPUT:

A vector with entries possibly in an extension of the base ring. This vector is an eigenvector for all Hecke operators acting via their transpose.

If lift = False, instead return an eigenvector in the subspace for the Hecke operators on the dual space. I.e., this is an eigenvector for the restrictions of Hecke operators to the dual space.

Note

  1. The answer is cached so subsequent calls always return the same vector. However, the algorithm is randomized, so calls during another session may yield a different eigenvector. This function is used mainly for computing systems of Hecke eigenvalues.

  2. One can also view a dual eigenvector as defining (via dot product) a functional phi from the ambient space of modular symbols to a field. This functional phi is an eigenvector for the dual action of Hecke operators on functionals.

EXAMPLES:

sage: SF = ModularSymbols(14).cuspidal_subspace().simple_factors()
sage: sorted([u.dual_eigenvector() for u in SF])
[(0, 1, 0, 0, 0), (1, 0, -3, 2, -1)]

>>> from sage.all import *
>>> SF = ModularSymbols(Integer(14)).cuspidal_subspace().simple_factors()
>>> sorted([u.dual_eigenvector() for u in SF])
[(0, 1, 0, 0, 0), (1, 0, -3, 2, -1)]
dual_hecke_matrix(n)[source]

Return the matrix of the \(n\)-th Hecke operator acting on the dual embedded representation of self.

EXAMPLES:

sage: CuspForms(1, 24).dual_hecke_matrix(5)
[     44656110        -15040]
[-307849789440      28412910]

>>> from sage.all import *
>>> CuspForms(Integer(1), Integer(24)).dual_hecke_matrix(Integer(5))
[     44656110        -15040]
[-307849789440      28412910]
eigenvalue(n, name='alpha')[source]

Assuming that self is a simple space, return the eigenvalue of the \(n\)-th Hecke operator on self.

INPUT:

  • n – index of Hecke operator

  • name – print representation of generator of eigenvalue field

EXAMPLES:

sage: A = ModularSymbols(125,sign=1).new_subspace()[0]
sage: A.eigenvalue(7)
-3
sage: A.eigenvalue(3)
-alpha - 2
sage: A.eigenvalue(3,'w')
-w - 2
sage: A.eigenvalue(3,'z').charpoly('x')
x^2 + 3*x + 1
sage: A.hecke_polynomial(3)
x^2 + 3*x + 1

>>> from sage.all import *
>>> A = ModularSymbols(Integer(125),sign=Integer(1)).new_subspace()[Integer(0)]
>>> A.eigenvalue(Integer(7))
-3
>>> A.eigenvalue(Integer(3))
-alpha - 2
>>> A.eigenvalue(Integer(3),'w')
-w - 2
>>> A.eigenvalue(Integer(3),'z').charpoly('x')
x^2 + 3*x + 1
>>> A.hecke_polynomial(Integer(3))
x^2 + 3*x + 1


sage: M = ModularSymbols(Gamma1(17)).decomposition()[8].plus_submodule()
sage: M.eigenvalue(2,'a')
a
sage: M.eigenvalue(4,'a')
4/3*a^3 + 17/3*a^2 + 28/3*a + 8/3

>>> from sage.all import *
>>> M = ModularSymbols(Gamma1(Integer(17))).decomposition()[Integer(8)].plus_submodule()
>>> M.eigenvalue(Integer(2),'a')
a
>>> M.eigenvalue(Integer(4),'a')
4/3*a^3 + 17/3*a^2 + 28/3*a + 8/3

Note

  1. In fact there are \(d\) systems of eigenvalues associated to self, where \(d\) is the rank of self. Each of the systems of eigenvalues is conjugate over the base field. This function chooses one of the systems and consistently returns eigenvalues from that system. Thus these are the coefficients \(a_n\) for \(n\geq 1\) of a modular eigenform attached to self.

  2. This function works even for Eisenstein subspaces, though it will not give the constant coefficient of one of the corresponding Eisenstein series (i.e., the generalized Bernoulli number).

factor_number()[source]

If this Hecke module was computed via a decomposition of another Hecke module, this is the corresponding number. Otherwise return -1.

EXAMPLES:

sage: ModularSymbols(23)[0].factor_number()
0
sage: ModularSymbols(23).factor_number()
-1

>>> from sage.all import *
>>> ModularSymbols(Integer(23))[Integer(0)].factor_number()
0
>>> ModularSymbols(Integer(23)).factor_number()
-1
gen(n)[source]

Return the \(n\)-th basis vector of the space.

EXAMPLES:

sage: ModularSymbols(23).gen(1)
(1,17)

>>> from sage.all import *
>>> ModularSymbols(Integer(23)).gen(Integer(1))
(1,17)
gens()[source]

Return a tuple of basis elements of self.

EXAMPLES:

sage: ModularSymbols(23).gens()
((1,0), (1,17), (1,19), (1,20), (1,21))

>>> from sage.all import *
>>> ModularSymbols(Integer(23)).gens()
((1,0), (1,17), (1,19), (1,20), (1,21))
hecke_matrix(n)[source]

Return the matrix of the \(n\)-th Hecke operator acting on given basis.

EXAMPLES:

sage: C = CuspForms(1, 16)
sage: C.hecke_matrix(3)
[-3348]

>>> from sage.all import *
>>> C = CuspForms(Integer(1), Integer(16))
>>> C.hecke_matrix(Integer(3))
[-3348]
hecke_operator(n)[source]

Return the \(n\)-th Hecke operator \(T_n\).

INPUT:

  • n – integer at least 1

EXAMPLES:

sage: M = ModularSymbols(11,2)
sage: T = M.hecke_operator(3) ; T
Hecke operator T_3 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: T.matrix()
[ 4  0 -1]
[ 0 -1  0]
[ 0  0 -1]
sage: T(M.0)
4*(1,0) - (1,9)
sage: S = M.cuspidal_submodule()
sage: T = S.hecke_operator(3) ; T
Hecke operator T_3 on Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: T.matrix()
[-1  0]
[ 0 -1]
sage: T(S.0)
-(1,8)

>>> from sage.all import *
>>> M = ModularSymbols(Integer(11),Integer(2))
>>> T = M.hecke_operator(Integer(3)) ; T
Hecke operator T_3 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
>>> T.matrix()
[ 4  0 -1]
[ 0 -1  0]
[ 0  0 -1]
>>> T(M.gen(0))
4*(1,0) - (1,9)
>>> S = M.cuspidal_submodule()
>>> T = S.hecke_operator(Integer(3)) ; T
Hecke operator T_3 on Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
>>> T.matrix()
[-1  0]
[ 0 -1]
>>> T(S.gen(0))
-(1,8)
hecke_polynomial(n, var='x')[source]

Return the characteristic polynomial of the \(n\)-th Hecke operator acting on this space.

INPUT:

  • n – integer

OUTPUT: a polynomial

EXAMPLES:

sage: ModularSymbols(11,2).hecke_polynomial(3)
x^3 - 2*x^2 - 7*x - 4

>>> from sage.all import *
>>> ModularSymbols(Integer(11),Integer(2)).hecke_polynomial(Integer(3))
x^3 - 2*x^2 - 7*x - 4
is_simple()[source]

Return True if this space is simple as a module for the corresponding Hecke algebra.

This raises NotImplementedError, as this is an abstract base class.

EXAMPLES:

sage: sage.modular.hecke.module.HeckeModule_free_module(QQ, 10, 3).is_simple()
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> sage.modular.hecke.module.HeckeModule_free_module(QQ, Integer(10), Integer(3)).is_simple()
Traceback (most recent call last):
...
NotImplementedError
is_splittable()[source]

Return True if and only if only it is possible to split off a nontrivial generalized eigenspace of self as the kernel of some Hecke operator (not necessarily prime to the level).

Note that the direct sum of several copies of the same simple module is not splittable in this sense.

EXAMPLES:

sage: M = ModularSymbols(Gamma0(64)).cuspidal_subspace()
sage: M.is_splittable()
True
sage: M.simple_factors()[0].is_splittable()
False

>>> from sage.all import *
>>> M = ModularSymbols(Gamma0(Integer(64))).cuspidal_subspace()
>>> M.is_splittable()
True
>>> M.simple_factors()[Integer(0)].is_splittable()
False
is_splittable_anemic()[source]

Return True if and only if only it is possible to split off a nontrivial generalized eigenspace of self as the kernel of some Hecke operator of index coprime to the level.

Note that the direct sum of several copies of the same simple module is not splittable in this sense.

EXAMPLES:

sage: M = ModularSymbols(Gamma0(64)).cuspidal_subspace()
sage: M.is_splittable_anemic()
True
sage: M.simple_factors()[0].is_splittable_anemic()
False

>>> from sage.all import *
>>> M = ModularSymbols(Gamma0(Integer(64))).cuspidal_subspace()
>>> M.is_splittable_anemic()
True
>>> M.simple_factors()[Integer(0)].is_splittable_anemic()
False
is_submodule(other)[source]

Return True if self is a submodule of other.

EXAMPLES:

sage: M = ModularSymbols(Gamma0(64))
sage: M[0].is_submodule(M)
True
sage: CuspForms(1, 24).is_submodule(ModularForms(1, 24))
True
sage: CuspForms(1, 12).is_submodule(CuspForms(3, 12))
False

>>> from sage.all import *
>>> M = ModularSymbols(Gamma0(Integer(64)))
>>> M[Integer(0)].is_submodule(M)
True
>>> CuspForms(Integer(1), Integer(24)).is_submodule(ModularForms(Integer(1), Integer(24)))
True
>>> CuspForms(Integer(1), Integer(12)).is_submodule(CuspForms(Integer(3), Integer(12)))
False
ngens()[source]

Return the number of generators of self.

This is equal to the rank.

EXAMPLES:

sage: ModularForms(1, 12).ngens()
2

>>> from sage.all import *
>>> ModularForms(Integer(1), Integer(12)).ngens()
2
projection()[source]

Return the projection map from the ambient space to self.

ALGORITHM: Let \(B\) be the matrix whose columns are obtained by concatenating together a basis for the factors of the ambient space. Then the projection matrix onto self is the submatrix of \(B^{-1}\) obtained from the rows corresponding to self, i.e., if the basis vectors for self appear as columns \(n\) through \(m\) of \(B\), then the projection matrix is got from rows \(n\) through \(m\) of \(B^{-1}\). This is because projection with respect to the B basis is just given by an \(m-n+1\) row slice \(P\) of a diagonal matrix D with 1s in the \(n\) through \(m\) positions, so projection with respect to the standard basis is given by \(P\cdot B^{-1}\), which is just rows \(n\) through \(m\) of \(B^{-1}\).

EXAMPLES:

sage: e = EllipticCurve('34a')
sage: m = ModularSymbols(34); s = m.cuspidal_submodule()
sage: d = s.decomposition(7)
sage: d
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(34) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 9 for Gamma_0(34) of weight 2 with sign 0 over Rational Field
]
sage: a = d[0]; a
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(34) of weight 2 with sign 0 over Rational Field
sage: pi = a.projection()
sage: pi(m([0,oo]))
-1/6*(2,7) + 1/6*(2,13) - 1/6*(2,31) + 1/6*(2,33)
sage: M = ModularSymbols(53,sign=1)
sage: S = M.cuspidal_subspace()[1] ; S
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 5 for Gamma_0(53) of weight 2 with sign 1 over Rational Field
sage: p = S.projection()
sage: S.basis()
((1,43) - (1,45), (1,47), (1,50))
sage: [ p(x) for x in S.basis() ]
[(1,43) - (1,45), (1,47), (1,50)]
sage: all(p(x)==x for x in S.basis())
True

>>> from sage.all import *
>>> e = EllipticCurve('34a')
>>> m = ModularSymbols(Integer(34)); s = m.cuspidal_submodule()
>>> d = s.decomposition(Integer(7))
>>> d
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(34) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 9 for Gamma_0(34) of weight 2 with sign 0 over Rational Field
]
>>> a = d[Integer(0)]; a
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(34) of weight 2 with sign 0 over Rational Field
>>> pi = a.projection()
>>> pi(m([Integer(0),oo]))
-1/6*(2,7) + 1/6*(2,13) - 1/6*(2,31) + 1/6*(2,33)
>>> M = ModularSymbols(Integer(53),sign=Integer(1))
>>> S = M.cuspidal_subspace()[Integer(1)] ; S
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 5 for Gamma_0(53) of weight 2 with sign 1 over Rational Field
>>> p = S.projection()
>>> S.basis()
((1,43) - (1,45), (1,47), (1,50))
>>> [ p(x) for x in S.basis() ]
[(1,43) - (1,45), (1,47), (1,50)]
>>> all(p(x)==x for x in S.basis())
True
system_of_eigenvalues(n, name='alpha')[source]

Assuming that self is a simple space of modular symbols, return the eigenvalues \([a_1, \ldots, a_nmax]\) of the Hecke operators on self. See self.eigenvalue(n) for more details.

INPUT:

  • n – number of eigenvalues

  • alpha – name of generate for eigenvalue field

EXAMPLES:

The outputs of the following tests are very unstable. The algorithms are randomized and depend on cached results. A slight change in the sequence of pseudo-random numbers or a modification in caching is likely to modify the results. We reset the random number generator and clear some caches for reproducibility:

sage: set_random_seed(0)
sage: ModularSymbols_clear_cache()

>>> from sage.all import *
>>> set_random_seed(Integer(0))
>>> ModularSymbols_clear_cache()

We compute eigenvalues for newforms of level 62:

sage: M = ModularSymbols(62,2,sign=-1)
sage: S = M.cuspidal_submodule().new_submodule()
sage: [[o.minpoly() for o in A.system_of_eigenvalues(3)] for A in S.decomposition()]
[[x - 1, x - 1, x], [x - 1, x + 1, x^2 - 2*x - 2]]

>>> from sage.all import *
>>> M = ModularSymbols(Integer(62),Integer(2),sign=-Integer(1))
>>> S = M.cuspidal_submodule().new_submodule()
>>> [[o.minpoly() for o in A.system_of_eigenvalues(Integer(3))] for A in S.decomposition()]
[[x - 1, x - 1, x], [x - 1, x + 1, x^2 - 2*x - 2]]

Next we define a function that does the above:

sage: def b(N, k=2):
....:    S = ModularSymbols(N,k,sign=-1).cuspidal_submodule().new_submodule()
....:    for A in S.decomposition():
....:        print("{} {}".format(N, A.system_of_eigenvalues(5)))

>>> from sage.all import *
>>> def b(N, k=Integer(2)):
...    S = ModularSymbols(N,k,sign=-Integer(1)).cuspidal_submodule().new_submodule()
...    for A in S.decomposition():
...        print("{} {}".format(N, A.system_of_eigenvalues(Integer(5))))


sage: b(63)
63 [1, 1, 0, -1, 2]
63 [1, alpha, 0, 1, -2*alpha]

>>> from sage.all import *
>>> b(Integer(63))
63 [1, 1, 0, -1, 2]
63 [1, alpha, 0, 1, -2*alpha]

This example illustrates finding field over which the eigenvalues are defined:

sage: M = ModularSymbols(23,2,sign=1).cuspidal_submodule().new_submodule()
sage: v = M.system_of_eigenvalues(10); v
[1, alpha, -2*alpha - 1, -alpha - 1, 2*alpha, alpha - 2, 2*alpha + 2, -2*alpha - 1, 2, -2*alpha + 2]
sage: v[0].parent()
Number Field in alpha with defining polynomial x^2 + x - 1

>>> from sage.all import *
>>> M = ModularSymbols(Integer(23),Integer(2),sign=Integer(1)).cuspidal_submodule().new_submodule()
>>> v = M.system_of_eigenvalues(Integer(10)); v
[1, alpha, -2*alpha - 1, -alpha - 1, 2*alpha, alpha - 2, 2*alpha + 2, -2*alpha - 1, 2, -2*alpha + 2]
>>> v[Integer(0)].parent()
Number Field in alpha with defining polynomial x^2 + x - 1

This example illustrates setting the print name of the eigenvalue field.

sage: A = ModularSymbols(125,sign=1).new_subspace()[0]
sage: A.system_of_eigenvalues(10)
[1, alpha, -alpha - 2, -alpha - 1, 0, -alpha - 1, -3, -2*alpha - 1, 3*alpha + 2, 0]
sage: A.system_of_eigenvalues(10,'x')
[1, x, -x - 2, -x - 1, 0, -x - 1, -3, -2*x - 1, 3*x + 2, 0]

>>> from sage.all import *
>>> A = ModularSymbols(Integer(125),sign=Integer(1)).new_subspace()[Integer(0)]
>>> A.system_of_eigenvalues(Integer(10))
[1, alpha, -alpha - 2, -alpha - 1, 0, -alpha - 1, -3, -2*alpha - 1, 3*alpha + 2, 0]
>>> A.system_of_eigenvalues(Integer(10),'x')
[1, x, -x - 2, -x - 1, 0, -x - 1, -3, -2*x - 1, 3*x + 2, 0]
weight()[source]

Return the weight of this Hecke module.

INPUT:

  • self – an arbitrary Hecke module

OUTPUT: integer; the weight

EXAMPLES:

sage: m = ModularSymbols(20, weight=2)
sage: m.weight()
2

>>> from sage.all import *
>>> m = ModularSymbols(Integer(20), weight=Integer(2))
>>> m.weight()
2
zero_submodule()[source]

Return the zero submodule of self.

EXAMPLES:

sage: ModularSymbols(11,4).zero_submodule()
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 6 for Gamma_0(11) of weight 4 with sign 0 over Rational Field
sage: CuspForms(11,4).zero_submodule()
Modular Forms subspace of dimension 0 of Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(11) of weight 4 over Rational Field

>>> from sage.all import *
>>> ModularSymbols(Integer(11),Integer(4)).zero_submodule()
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 6 for Gamma_0(11) of weight 4 with sign 0 over Rational Field
>>> CuspForms(Integer(11),Integer(4)).zero_submodule()
Modular Forms subspace of dimension 0 of Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(11) of weight 4 over Rational Field
class sage.modular.hecke.module.HeckeModule_generic(base_ring, level, category=None)[source]

Bases: Module

A very general base class for Hecke modules.

We define a Hecke module of weight \(k\) to be a module over a commutative ring equipped with an action of operators \(T_m\) for all positive integers \(m\) coprime to some integer \(n`(the level), which satisfy `T_r T_s = T_{rs}\) for \(r,s\) coprime, and for powers of a prime \(p\), \(T_{p^r} = T_{p} T_{p^{r-1}} - \varepsilon(p) p^{k-1} T_{p^{r-2}}\), where \(\varepsilon(p)\) is some endomorphism of the module which commutes with the \(T_m\).

We distinguish between full Hecke modules, which also have an action of operators \(T_m\) for \(m\) not assumed to be coprime to the level, and anemic Hecke modules, for which this does not hold.

Element[source]

alias of HeckeModuleElement

anemic_hecke_algebra()[source]

Return the Hecke algebra associated to this Hecke module.

EXAMPLES:

sage: T = ModularSymbols(1,12).hecke_algebra()
sage: A = ModularSymbols(1,12).anemic_hecke_algebra()
sage: T == A
False
sage: A
Anemic Hecke algebra acting on Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
sage: A.is_anemic()
True

>>> from sage.all import *
>>> T = ModularSymbols(Integer(1),Integer(12)).hecke_algebra()
>>> A = ModularSymbols(Integer(1),Integer(12)).anemic_hecke_algebra()
>>> T == A
False
>>> A
Anemic Hecke algebra acting on Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
>>> A.is_anemic()
True
character()[source]

Return the character of this space.

As this is an abstract base class, return None.

EXAMPLES:

sage: sage.modular.hecke.module.HeckeModule_generic(QQ, 10).character() is None
True

>>> from sage.all import *
>>> sage.modular.hecke.module.HeckeModule_generic(QQ, Integer(10)).character() is None
True
dimension()[source]

Synonym for rank().

EXAMPLES:

sage: M = sage.modular.hecke.module.HeckeModule_generic(QQ, 10).dimension()
Traceback (most recent call last):
...
NotImplementedError: Derived subclasses must implement rank

>>> from sage.all import *
>>> M = sage.modular.hecke.module.HeckeModule_generic(QQ, Integer(10)).dimension()
Traceback (most recent call last):
...
NotImplementedError: Derived subclasses must implement rank
hecke_algebra()[source]

Return the Hecke algebra associated to this Hecke module.

EXAMPLES:

sage: T = ModularSymbols(Gamma1(5),3).hecke_algebra()
sage: T
Full Hecke algebra acting on Modular Symbols space of dimension 4 for Gamma_1(5) of weight 3 with sign 0 over Rational Field
sage: T.is_anemic()
False

>>> from sage.all import *
>>> T = ModularSymbols(Gamma1(Integer(5)),Integer(3)).hecke_algebra()
>>> T
Full Hecke algebra acting on Modular Symbols space of dimension 4 for Gamma_1(5) of weight 3 with sign 0 over Rational Field
>>> T.is_anemic()
False


sage: M = ModularSymbols(37,sign=1)
sage: E, A, B = M.decomposition()
sage: A.hecke_algebra() == B.hecke_algebra()
False

>>> from sage.all import *
>>> M = ModularSymbols(Integer(37),sign=Integer(1))
>>> E, A, B = M.decomposition()
>>> A.hecke_algebra() == B.hecke_algebra()
False
is_full_hecke_module()[source]

Return True if this space is invariant under all Hecke operators.

Since self is guaranteed to be an anemic Hecke module, the significance of this function is that it also ensures invariance under Hecke operators of index that divide the level.

EXAMPLES:

sage: M = ModularSymbols(22); M.is_full_hecke_module()
True
sage: M.submodule(M.free_module().span([M.0.list()]), check=False).is_full_hecke_module()
False

>>> from sage.all import *
>>> M = ModularSymbols(Integer(22)); M.is_full_hecke_module()
True
>>> M.submodule(M.free_module().span([M.gen(0).list()]), check=False).is_full_hecke_module()
False
is_hecke_invariant(n)[source]

Return True if self is invariant under the Hecke operator \(T_n\).

Since self is guaranteed to be an anemic Hecke module it is only interesting to call this function when \(n\) is not coprime to the level.

EXAMPLES:

sage: M = ModularSymbols(22).cuspidal_subspace()
sage: M.is_hecke_invariant(2)
True

>>> from sage.all import *
>>> M = ModularSymbols(Integer(22)).cuspidal_subspace()
>>> M.is_hecke_invariant(Integer(2))
True

We use check=False to create a nasty “module” that is not invariant under \(T_2\):

sage: S = M.submodule(M.free_module().span([M.0.list()]), check=False); S
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 7 for Gamma_0(22) of weight 2 with sign 0 over Rational Field
sage: S.is_hecke_invariant(2)
False
sage: [n for n in range(1,12) if S.is_hecke_invariant(n)]
[1, 3, 5, 7, 9, 11]

>>> from sage.all import *
>>> S = M.submodule(M.free_module().span([M.gen(0).list()]), check=False); S
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 7 for Gamma_0(22) of weight 2 with sign 0 over Rational Field
>>> S.is_hecke_invariant(Integer(2))
False
>>> [n for n in range(Integer(1),Integer(12)) if S.is_hecke_invariant(n)]
[1, 3, 5, 7, 9, 11]
is_zero()[source]

Return True if this Hecke module has dimension 0.

EXAMPLES:

sage: ModularSymbols(11).is_zero()
False
sage: ModularSymbols(11).old_submodule().is_zero()
True
sage: CuspForms(10).is_zero()
True
sage: CuspForms(1,12).is_zero()
False

>>> from sage.all import *
>>> ModularSymbols(Integer(11)).is_zero()
False
>>> ModularSymbols(Integer(11)).old_submodule().is_zero()
True
>>> CuspForms(Integer(10)).is_zero()
True
>>> CuspForms(Integer(1),Integer(12)).is_zero()
False
level()[source]

Return the level of this modular symbols space.

INPUT:

  • ModularSymbols self – an arbitrary space of modular symbols

OUTPUT: integer; the level

EXAMPLES:

sage: m = ModularSymbols(20)
sage: m.level()
20

>>> from sage.all import *
>>> m = ModularSymbols(Integer(20))
>>> m.level()
20
rank()[source]

Return the rank of this module over its base ring.

This raises a NotImplementedError, since this is an abstract base class.

EXAMPLES:

sage: sage.modular.hecke.module.HeckeModule_generic(QQ, 10).rank()
Traceback (most recent call last):
...
NotImplementedError: Derived subclasses must implement rank

>>> from sage.all import *
>>> sage.modular.hecke.module.HeckeModule_generic(QQ, Integer(10)).rank()
Traceback (most recent call last):
...
NotImplementedError: Derived subclasses must implement rank
submodule(X)[source]

Return the submodule of self corresponding to X.

As this is an abstract base class, this raises a NotImplementedError.

EXAMPLES:

sage: sage.modular.hecke.module.HeckeModule_generic(QQ, 10).submodule(0)
Traceback (most recent call last):
...
NotImplementedError: Derived subclasses should implement submodule

>>> from sage.all import *
>>> sage.modular.hecke.module.HeckeModule_generic(QQ, Integer(10)).submodule(Integer(0))
Traceback (most recent call last):
...
NotImplementedError: Derived subclasses should implement submodule
sage.modular.hecke.module.is_HeckeModule(x)[source]

Return True if x is a Hecke module.

EXAMPLES:

sage: from sage.modular.hecke.module import is_HeckeModule
sage: is_HeckeModule(ModularForms(Gamma0(7), 4))
doctest:warning...
DeprecationWarning: the function is_HeckeModule is deprecated;
use 'isinstance(..., HeckeModule_generic)' instead
See https://github.com/sagemath/sage/issues/37895 for details.
True
sage: is_HeckeModule(QQ^3)
False
sage: is_HeckeModule(J0(37).homology())
True

>>> from sage.all import *
>>> from sage.modular.hecke.module import is_HeckeModule
>>> is_HeckeModule(ModularForms(Gamma0(Integer(7)), Integer(4)))
doctest:warning...
DeprecationWarning: the function is_HeckeModule is deprecated;
use 'isinstance(..., HeckeModule_generic)' instead
See https://github.com/sagemath/sage/issues/37895 for details.
True
>>> is_HeckeModule(QQ**Integer(3))
False
>>> is_HeckeModule(J0(Integer(37)).homology())
True