Operators

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

Compute a matrix representing the \(n\)th Hecke operator \(T_n\) with respect to Basis(M) for the supersingular module \(M\).

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

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

Example: Operators (ex-d33253)

In this example we observe that \(T_2\) and \(W_3\) have the same characteristic polynomial on \(S_2(\Gamma_0(33))\) as on the cuspidal subspace of the supersingular module with \(p=11\), \(N=3\).

> 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)));
[
    <$.1 - 1, 1>,
    <$.1 + 1, 2>
]
> Factorization(CharacteristicPolynomial(AtkinLehnerOperator(MF, 3)));
[
    <$.1 - 1, 2>,
    <$.1 + 1, 1>
]

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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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