Compute a matrix representing the nth Hecke operator Tn with respect to Basis(M) for the supersingular module M.
A matrix representing the Atkin-Lehner involution Wq on the supersingular module M. The number q must equal either Prime(M) or AuxiliaryLevel(M).
> 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]