# Attributes

## `BaseField(M): ModFrmBianchi -> FldNum`

## `BaseRing(M): ModFrmBianchi -> FldNum`

## `CoefficientField(M): ModFrmBianchi -> FldNum`

## `CoefficientRing(M): ModFrmBianchi -> FldNum`

The field on which the space $M$ of Bianchi modular forms was defined.

## `Level(M): ModFrmBianchi -> RngOrdIdl`

The level of the space $M$.

## `Dimension(M): ModFrmBianchi -> RngIntElt`

The dimension of the space $M$. Dimension formulas are not available, so the dimension is computed by explicit construction of the space.

## `VoronoiData(M): ModFrmBianchi -> Rec`

This returns a record containing technical data that is computed in the precomputation phase of the algorithm. This depends only on the base field of $M$, and the data can be reused when computing spaces of different levels over the same field.

## `Example: Creation Example (ex-1ddde9)`

We create spaces of modular forms over ${\mathbb{Q}}(\sqrt{-14})$ for various levels.

```magma
> _<x> := PolynomialRing(Rationals());
> F := NumberField(x^2 + 14);
> OF := Integers(F);
> level := 1*OF;
> M := BianchiCuspForms(F, level);
> M;
Cuspidal space of Bianchi modular forms over
    Number Field with defining polynomial x^2 + 14 over the Rational Field
    Level = Ideal of norm 1 generated by ( [1, 0] )
    Weight = 2
>  time Dimension(M);
0
Time: 0.050

```

We now define a space with level equal to the square of one of the split primes dividing $3$.

```magma
> level := (Factorization(3*OF)[1][1])^2;
> Norm(level);
9
> time M9 := BianchiCuspForms(F, level);
Time: 1.370
> time Dimension(M9);
1
Time: 0.370

```

When defining this space, we may tell Magma to use the same Voronoi data, to avoid repeating this expensive precomputation:

```magma
> time M9 := BianchiCuspForms(F, level : VorData := VoronoiData(M) );
Time: 0.000
> time Dimension(M9);
1
Time: 0.370
> M9;
Cuspidal space of Bianchi modular forms over
    Number Field with defining polynomial z^2 + 14 over the Rational Field
    Level = Ideal of norm 9 generated by ( [9, 0], [5, 2] )
    Weight = 2
    Dimension 1

```