26.7 Hecke algebras and modules

Module: sage.modular.hecke.algebra

Module-level Functions

AnemicHeckeAlgebra( M)

HeckeAlgebra( M)

is_HeckeAlgebra( x)

Class: HeckeAlgebra_anemic

class HeckeAlgebra_anemic

Functions: gens,$  $ hecke_operator,$  $ is_anemic

gens( self)

Return a generator over all Hecke operator $ T_n$ for $ n = 1, 2, 3, \ldots$ , with $ n$ coprime to the level. This is an infinite sequence.

sage: T = ModularSymbols(12,2).anemic_hecke_algebra()
sage: g = T.gens()
sage: g.next()
Hecke operator T_1 on Modular Symbols space of dimension 5 for Gamma_0(12)
of weight 2 with sign 0 over Rational Field
sage: g.next()
Hecke operator T_5 on Modular Symbols space of dimension 5 for Gamma_0(12)
of weight 2 with sign 0 over Rational Field

hecke_operator( self, n)

Return the $ n$ -th Hecke operator, for $ n$ any positive integer coprime to the level.

sage: T = ModularSymbols(Gamma1(5),3).anemic_hecke_algebra()
sage: T.hecke_operator(2)
Hecke operator T_2 on Modular Symbols space of dimension 4 for Gamma_1(5)
of weight 3 with sign 0 and over Rational Field
sage: T.hecke_operator(5)
Traceback (most recent call last):
...
IndexError: Hecke operator T_5 not defined in the anemic Hecke algebra

Special Functions: __cmp__,$  $ _latex_,$  $ _repr_

Class: HeckeAlgebra_base

class HeckeAlgebra_base
An algebra of Hecke operators on a fixed Hecke module
HeckeAlgebra_base( self, M)

INPUT:
    M -- a Hecke module

Functions: basis,$  $ discriminant,$  $ gen,$  $ gens,$  $ hecke_matrix,$  $ hecke_operator,$  $ is_noetherian,$  $ level,$  $ module,$  $ ngens,$  $ rank

gen( self, n)

Return the $ n$ -th Hecke operator.

sage: T = ModularSymbols(11).hecke_algebra()
sage: T.gen(2)
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(11)
of weight 2 with sign 0 over Rational Field

gens( self)

Return a generator over all Hecke operator $ T_n$ for $ n = 1, 2, 3, \ldots$ . This is infinite.

sage: T = ModularSymbols(1,12).hecke_algebra()
sage: g = T.gens()
sage: g.next()
Hecke operator T_1 on Modular Symbols space of dimension 3 for Gamma_0(1)
of weight 12 with sign 0 over Rational Field
sage: g.next()
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1)
of weight 12 with sign 0 over Rational Field

hecke_matrix( self, n)

Return the matrix of the n-th Hecke operator $ T_n$ .

sage: T = ModularSymbols(1,12).hecke_algebra()
sage: T.hecke_matrix(2)
[ -24    0    0]
[   0  -24    0]
[4860    0 2049]

hecke_operator( self, n)

Return the n-th Hecke operator $ T_n$ .

sage: T = ModularSymbols(1,12).hecke_algebra()
sage: T.hecke_operator(2)
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1)
of weight 12 with sign 0 over Rational Field

is_noetherian( self)

Return True if this Hecke algebra is Noetherian as a ring.

module( self)

sage: T = ModularSymbols(1,12).hecke_algebra()
sage: T.module()
Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign
0 over Rational Field

Special Functions: __call__,$  $ __contains__,$  $ _HeckeAlgebra_base__matrix_space,$  $ _latex_,$  $ _repr_

__contains__( self, x)

sage: T = ModularSymbols(11).hecke_algebra()
sage: T.gen(2) in T
True
sage: 5 in T
False

Class: HeckeAlgebra_full

class HeckeAlgebra_full

Functions: is_anemic

is_anemic( self)

Return True if this is an anemic Hecke algebra.

Special Functions: __cmp__,$  $ _repr_

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