I am currently going through Shimura's paper on half-integer weight modular forms. I would like to understand given a -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!



Computations of half-integer forms in SAGE/Magma
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](https://pari.math.u-bordeaux.fr/doc.html) (see the section *Modular forms*). If you want to know the mathematics behind the algorithms, you can read [this article](https://arxiv.org/abs/1810.00547) by Belabas and Cohen.


