Example ModSS_Operators (H144E8)

In this example we observe that T₂ and W₃ have the same characteristic polynomial on S₂(Γ₀(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>
]

The supersingular module with p=3 and N=11 is isomorphic as a module to the subspace of 3-new cuspforms in S₂(Γ₀(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]