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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!

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  • $\begingroup$ I'd be very surprised if this were implemented, it's already pretty difficult to compute the action of Atkin--Lehner operators in integer weight (there are no simple formulas for the action in terms of q-expansions). $\endgroup$ Commented Jul 5, 2023 at 12:52

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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.

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  • $\begingroup$ Thank you very much! I will look into it. I had some idea about Magma/Sage that is why I was trying to compute it by using either of these. I do not have any idea about PARI/GP. I will try to use the manual of it to use the suggestions you provided. Thank you again. $\endgroup$ Commented Jul 17, 2023 at 1:31
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    $\begingroup$ Maybe the tutorial on modular forms can be helpful: pari.math.u-bordeaux.fr/pub/pari/manuals/2.15.1/tutorial-mf.pdf $\endgroup$ Commented Jul 18, 2023 at 6:33

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