# 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 $p=11$, $N=3$.

```magma
> 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>
]

```

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

```magma
> 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]

```