11.3.1.2 HeckeAlgebra_base Objects

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

INPUT:
    M -- a Hecke module

Instances of class HeckeAlgebra_base have the following methods (in addition to inherited methods and special methods):

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

Further documentation:

gen( n)

Return the $ n$ -th Hecke operator.

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

gens( )

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 Full Modular Symbols space for Gamma_0(1) of weight
12 with sign 0 and dimension 3 over Rational Field
sage: g.next()
Hecke operator T_2 on Full Modular Symbols space for Gamma_0(1) of weight
12 with sign 0 and dimension 3 over Rational Field

hecke_matrix( 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( 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 Full Modular Symbols space for Gamma_0(1) of weight
12 with sign 0 and dimension 3 over Rational Field

is_noetherian( )

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

module( )

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

Instances of class HeckeAlgebra_base also have the following special methods:

__call__,$  $ __contains__,$  $ __repr__,$  $ _HeckeAlgebra_base__matrix_space,$  $ _latex_

Further documentation:

__contains__( x)

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

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