Computations of half-integer forms in SAGE/Magma

Question

I am currently going through Shimura's paper on half-integer weight modular forms. I would like to understand given a 𝑞-expansion of half-integral weight modular forms of arbitrary level and character, how to compute the effect of the Hecke operator and the Atkin-Lehner operator/Fricke involution in SAGE/Magma. I am a beginner in SAGE/Magma and
The only documentation I found is related to computing basis of weight k/2 and character chi.

If anyone has any sources on or explanations as to how to compute how q-expansions are transformed under the action of Hecke operators and Atkin-Lehner/Fricke involution, it would be greatly appreciated!

Answer 1

This can be done using PARI/GP, which can deal with spaces of modular forms of half-integral weight. Given a modular form $f$ of weight $k$ (possibly half-integral), the command mfslashexpansion can compute the $q$-expansion of $f |_k g$ for any $g \in \mathrm{GL}_2^+(\mathbf{Q})$. This relies on a floating-point method but works well in practice. One of the ideas is to multiply your half-integral weight modular form by the weight $1/2$ modular form $\theta$, whose behaviour under the Atkin-Lehner involution is known. This reduces to the case of integral weight, which is still difficult, but one other idea is to write the form as a linear combination of pairwise products of Eisenstein series. The behaviour of Eisenstein series under the Atkin-Lehner involution is known, which gives the result.

You can look at PARI/GP's users manual (see the section Modular forms). If you want to know the mathematics behind the algorithms, you can read this article by Belabas and Cohen.