For 1-dimensional Galois orbits of cuspidal eigenforms in spaces of
ternary orthogonal modular forms, returns the associated
classical newform.
> ModularForm(fs[2]);
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11
+ O(q^12)
> fQ;
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11
+ O(q^12)
Theta1(f) : ModFrmAlgElt -> RngSerPowElt
Precision: RngIntElt Default: 25
The theta series associated to f, with precision qn,
where n is given by Precision.
Theta1 returns the normalized cuspidal newform.
Theta2(f) : ModFrmAlgElt -> Assoc
Precision: RngIntElt Default: 25
The theta series of genus g associated to f, which is
a Siegel modular form of genus g, given as an
associative array whose keys are the exponents ei, j
of the variables qi, j, and whose values are the
coefficient in the Fourier expansion of f for the
monomial qe = ∏i, j qi, jei, j.
Precision determines the maximal exponent ei, i
for the diagonal entries qi, i.
Theta2 is the same as setting g = 2.
Precision: RngIntElt Default: 25
The Shimura Lift of f in the space Mk(N) with precision qn,
where n is given by Precision.
The theta series of a ternary orthogonal modular form f
is not the same as the cuspidal modular form associated to it,
but one can relate these through the Shimura lift.
For example, we consider the space of orthogonal modular forms
of rank 3 and discriminant 11 2.
> L:= TernaryQuadraticLattices(121)[1][1];
> M := OrthogonalModularForms(L);
> fs := HeckeEigenforms(M);
> f := fs[2];
> theta<q> := Theta1(f : Precision := 25^2);
> theta + O(q^25);
q + q^3 - 2*q^4 - q^5 - q^11 - 2*q^14 + 3*q^15 + 2*q^16 + 2*q^20 + 2*q^22 -
3*q^23 + O(q^25)
> qExpansion(&+Basis(CuspForms(44,3/2)),25);
q + q^3 - 2*q^4 - q^5 - q^11 - 2*q^14 + 3*q^15 + 2*q^16 + 2*q^20 + 2*q^22 -
3*q^23 + O(q^25)
As expected θ(f) ∈S3/2(44), but using the Shimura lift, we can associate
to it a form in S2(11). Recall that the initial construction does not yield
a newform, as we see explicitly below.
> ShimuraLift(theta, 2, 11);
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 - 2*q^12 + 4*q^13
+ 4*q^14 - q^15 - 4*q^16 - 2*q^17 + 4*q^18 + 2*q^20 + 2*q^21 - q^23 +
O(q^25)
> f11 := qExpansion(Basis(CuspForms(11,2))[1],25);
> f11 - Evaluate(f11, q^11);
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 - 2*q^12 + 4*q^13
+ 4*q^14 - q^15 - 4*q^16 - 2*q^17 + 4*q^18 + 2*q^20 + 2*q^21 - q^23 +
O(q^25)
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|