# Hecke Operators

The computations are done essentially on the cohomology of $\Gamma \backslash \bf{H}_3$, and so the for a non-principal ideal $P$, the Hecke operator $T_P$ does not act on this space, but sometimes the Hecke action can be deduced from the action of principal Hecke operators. See for instance [[Lingham, 2005](../../references.md#cite-linghamthesis)]. Specifically, suppose ${\mathfrak{p}}$ is a prime ideal that is prime to the level. There exists an ideal ${\mathfrak{a}}$, prime to ${\mathfrak{p}}$ and the level, such that ${\mathfrak{a}}^2 {\mathfrak{p}}$ is principal. Then the composition $T_{{\mathfrak{a}},{\mathfrak{a}}}T_{{\mathfrak{p}}}$ 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 $T_P$. When $P$ is prime and a square in the class group, $T$ is the composition $T_{{\mathfrak{a}},{\mathfrak{a}}}T_P$ for a suitably chosen ideal ${\mathfrak{a}}$.

## `Example: hecke (ex-72d295)`

We continue the previous example.

```magma
> _<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.