11.3.2.3 HeckeOperator Objects

class HeckeOperator
HeckeOperator( parent, n)

sage: M = ModularSymbols(11)
sage: M.hecke_operator(2005)
Hecke operator T_2005 on Full Modular Symbols space for Gamma_0(11) of
weight 2 with sign 0 and dimension 3 over Rational Field

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

index,$  $ matrix,$  $ matrix_form

Further documentation:

index( )

Return the index of this Hecke operator, i.e., if this Hecke operator is $ T_n$ , return the int $ n$ .

sage: T = ModularSymbols(11).hecke_operator(17)
sage: T.index()
17

matrix( )

Return the matrix underlying this Hecke operator.

sage: T = ModularSymbols(11).hecke_operator(17)
sage: T.matrix()
[18  0 -4]
[ 0 -2  0]
[ 0  0 -2]

matrix_form( )

Return the matrix form of this element of a Hecke algebra.

sage: T = ModularSymbols(11).hecke_operator(17)
sage: T.matrix_form()
Hecke operator on Full Modular Symbols space for Gamma_0(11) of weight 2
with sign 0 and dimension 3 over Rational Field defined by:
[18  0 -4]
[ 0 -2  0]
[ 0  0 -2]

Instances of class HeckeOperator also have the following special methods:

__cmp__,$  $ __mul__,$  $ __repr__,$  $ _latex_

Further documentation:

__mul__( other)

We create the space of modular symbols of level $ 11$ and weight $ 2$ , then compute $ T_2$ and $ T_3$ on it, along with their composition.

sage: M = ModularSymbols(11)
sage: t2 = M.hecke_operator(2); t3 = M.hecke_operator(3)
sage: t2*t3
Hecke operator T_6 on Full Modular Symbols space for Gamma_0(11) of weight
2 with sign 0 and dimension 3 over Rational Field
sage: t3.matrix() * t2.matrix()
[12  0 -2]
[ 0  2  0]
[ 0  0  2]
sage: (t2*t3).matrix()
[12  0 -2]
[ 0  2  0]
[ 0  0  2]

When we compute $ T_2^2$ the result is not (easily seen to be) a Hecke operator of the form $ T_n$ , so it is returned as a Hecke module homomorphism defined as a matrix:

sage: t2**5
Hecke operator on Full Modular Symbols space for Gamma_0(11) of weight 2
with sign 0 and dimension 3 over Rational Field defined by:
[243   0 -55]
[  0 -32   0]
[  0   0 -32]

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