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.
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.
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.
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).
This is a list containing an eigenform from each space in NewformDecomposition(M).
Given a space M constructed using NewformDecomposition, this returns the number field over which the Eigenform of M is defined.
The computes the eigenvalue of the Hecke operator TP acting on the eigenform f (which should be a Hilbert modular form constructed using Eigenform).
> 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).
> 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 7Happily, they agree with the eigenvalues of the elliptic cusp form of conductor 11 over Q:
> 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 7The 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.
> K := HeckeEigenvalueField(decomp[2]); > K; Number Field with defining polynomial $.1^5 - 8*$.1^3 + 10*$.1 + 4 over F > IsTotallyReal(K); true