# Eigenspace Decomposition and Eigenforms

## `HeckeEigenvalueBound(M, P): ModFrmHil, RngOrdIdl -> RngIntElt`

Returns a bound on the absolute value of the Hecke eigenvalue at the prime $P$ which must hold for all newforms in the space $M$ of Hilbert modular forms.

## `NewformDecomposition(M): ModFrmHil -> List`

Given a space $M$ of Hilbert modular forms which was created as a `NewSubspace`, this decomposes $M$ into subspaces that are irreducible modules under the Hecke action.

## `NewformsOfDegree1(M): ModFrmHil -> List`

This constructs the list of new eigenforms in $M$ that have rational eigenvalues, i.e. corresponding to the $1$-dimensional components in the `NewformDecomposition`. The space $M$ is not required to be a new space. The algorithm avoids constructing the new subspace of $M$, and makes use of bounds on the eigenvalues.

## `Eigenform(M): ModFrmHil -> ModFrmHilElt`

This constructs an eigenform contained in the space $M$ of Hilbert modular forms (which should be an irreducible module under the Hecke action, for instance a space obtained using `NewformDecomposition`).

## `Eigenforms(M): ModFrmHil -> List`

This is a list containing an eigenform from each space in `NewformDecomposition(M)`.

## `HeckeEigenvalueField(M): ModFrmHil -> Fld`

Given a space $M$ constructed using `NewformDecomposition`, this returns the number field over which the `Eigenform` of $M$ is defined.

## `HeckeEigenvalue(f, P): ModFrmHilElt, RngOrdIdl -> FldAlgElt`

This computes the eigenvalue of the Hecke operator $T_P$ acting on the eigenform $f$ (which should be a Hilbert modular form constructed using `Eigenform`).

## `Example: Eigenform Examples (ex-15f603)`

We compute the newforms corresponding to elliptic curves over ${\mathbb{Q}}(\sqrt{2})$ of conductor $11$, and find there is only the one coming from ${\mathbb{Q}}$.

```magma
> R<x> := PolynomialRing(IntegerRing());
> F := NumberField(x^2-2);  OF := Integers(F);
> M := HilbertCuspForms(F, 11*OF);
> Dimension(M);
6
> time decomp := NewformDecomposition(NewSubspace(M)); decomp;
Time: 11.130
[*
    New cuspidal space of Hilbert modular forms of dimension 1 over
    Number Field with defining polynomial x^2 - 2 over the Rational Field
       Level = Ideal of norm 121 generated by ( [11, 0] )
       Weight = [ 2, 2 ],
    New cuspidal space of Hilbert modular forms of dimension 5 over
    Number Field with defining polynomial x^2 - 2 over the Rational Field
       Level = Ideal of norm 121 generated by ( [11, 0] )
       Weight = [ 2, 2 ]
*]

```

We look at the first few eigenvalues of the $1$-dimension piece (at split primes).

```magma
> f := Eigenform(decomp[1]);
> primes := [P : P in PrimesUpTo(40,F) | IsOdd(Norm(P)) and IsPrime(Norm(P))];
> for P in primes do
>   Norm(P),  HeckeEigenvalue(f,P);
> end for;
7 -2
7 -2
17 -2
17 -2
23 -1
23 -1
31 7
31 7

```

Happily, they agree with the eigenvalues of the elliptic cusp form of conductor $11$ over ${\mathbb{Q}}$:

```magma
> fQ := Newforms(CuspForms(11))[1][1];
> for P in primes do
>   p := Norm(P);
>   p,  Coefficient(fQ, p);
> end for;
7 -2
7 -2
17 -2
17 -2
23 -1
23 -1
31 7
31 7

```

The $5$-dimensional piece conjecturally corresponds to an abelian variety over $F$ of dimension $5$, which would be absolutely irreducible and have real multiplication by the following field.

```magma
> K := HeckeEigenvalueField(decomp[2]);
> K;
Number Field with defining polynomial $.1^5 - 8*$.1^3 + 10*$.1 + 4 over F
> IsTotallyReal(K);
true

```