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Hecke Operators





HeckeOperator(M, n) : ModSS, RngIntElt -> AlgMatElt



Compute a matrix representing the nth Hecke operator T_n 
with respect to Basis(M).



AtkinLehnerOperator(M, q) : ModSS, RngIntElt -> AlgMatElt



A matrix representing the Atkin-Lehner involution W_q on M.
The number q must equal either Prime(M) or 
AuxiliaryLevel(M).





Example ModSS_Hecke operators (H92E8)

We observe that T_2 and W_3 act with the same characteristic
polynomial on the cuspidal subspaces of the supersingular 
module with p=11, N=3 and the cuspforms S_2(Gamma_0(33)).





> SS := CuspidalSubspace(SupersingularModule(11,3));
> MF := CuspForms(33,2);
> Factorization(CharacteristicPolynomial(HeckeOperator(SS,2)));
[
    <.1 - 1, 1>,
    <.1 + 2, 2>
]
> Factorization(CharacteristicPolynomial(HeckeOperator(MF,2)));
[
    <.1 - 1, 1>,
    <.1 + 2, 2>
]
> Factorization(CharacteristicPolynomial(AtkinLehnerOperator(SS,3)));
[
    <.1 - 1, 2>,
    <.1 + 1, 1>
]
> Factorization(CharacteristicPolynomial(AtkinLehnerOperator(MF,3)));
[
    <.1 - 1, 2>,
    <.1 + 1, 1>
]


The supersingular module with p=3 and N=11 is isomorphic
as a module to the subspace of 3-new cuspforms 
in S_2(Gamma_0(33)).





> SS := CuspidalSubspace(SupersingularModule(3,11));
> MF := NewSubspace(CuspForms(33,2),3);
> HeckeOperator(SS,17);
[-2]
> HeckeOperator(MF,17);
[-2]
> AtkinLehnerOperator(SS,11);
[1]
> AtkinLehnerOperator(MF,11);
[1]





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