Relation to Other Modular Forms

ModularForm(f) : ModFrmAlgElt -> ModFrmElt
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)
ThetaSeries(f) : ModFrmAlgElt -> RngSerPowElt
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.
ThetaSiegel(f, g) : ModFrmAlgElt, RngIntElt -> Assoc
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.
ShimuraLift(f, k, N) : RngSerPowElt, RngIntElt, RngIntElt -> RngSerPowElt
    Precision: RngIntElt                Default: 25
The Shimura Lift of f in the space Mk(N) with precision qn, where n is given by Precision.

Example ModFrmAlg_shimura_lift (H151E5)

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 112. 
> 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)
V2.29, 21 October 2025