Hecke Operators

The computations are done essentially on the cohomology of Γ \ HH3, and so the for a non-principal ideal P, the Hecke operator TP does not act on this space, but sometimes the Hecke action can be deduced from the action of principal Hecke operators. See for instance [Lin05]. Specifically, suppose fp is a prime ideal that is prime to the level. There exists an ideal fa, prime to fp and the level, such that fa2 fp is principal. Then the composition Tfa, faTfp acts on the cohomology.

HeckeOperator(M, P) : ModFrmBianchi, RngOrdIdl -> Mtrx
This returns a matrix representing a certain Hecke action T on the space M of Bianchi modular forms, with respect to the fixed basis of M. The ideal P must be principal (but not necessarily prime) or prime and a square in the class group. The ideal must also be coprime to the level (except if the space is dimension 0). If P is principal, then T is the Hecke operator TP. When P is prime and a square in the class group, T is the composition Tfa, faTP for a suitably chosen ideal fa.

Example ModFrmBianchi_hecke (H147E2)

We continue the previous example. 
> _<x> := PolynomialRing(Rationals());
>  F := NumberField(x^2+14);
>  OF := Integers(F);
>  level := (Factorization(3*OF)[1][1])^2;
>  M9 := BianchiCuspForms(F, level);
>  P:=Factorization(23*OF);
>  P[1,1];
Prime Ideal of OF
Two element generators:
    [23, 0]
    [3, 1]
>  HeckeOperator(M9, P[1,1]);
[8]
>  P[2,1];
Prime Ideal of OF
Two element generators:
    [23, 0]
    [20, 1]
>  HeckeOperator(M9, P[2,1]);
[-8]
>  HeckeOperator(M9, 2*OF);
[1]
Since this cuspidal space has dimension 1, it consists of a single eigenform, whose eigenvalues can be read from the Hecke matrices.
V2.28, 13 July 2023