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Computational Aspects of Modular Forms and Galois Representations by Robin de Jong, Jean-Marc Couveignes, Franz Merkl, Johan Bosman, Bas Edixhoven

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Chapter Fifteen

Computing coefficients of modular forms

by Bas Edixhoven

 

In this chapter we apply our main result on the computation of Galois representations attached to modular forms of level one to the computation of coefficients of modular forms. In Section 15.1 we treat the case of the discriminant modular form, that is, the computation of Ramanujan’s τ-function at primes. In Section 15.2 we deal with the more general case of forms of level one and arbitrary weight k, reformulated as the computation of Hecke operators Tn as Image-linear combinations of the Ti with i < k/12. In Section 15.3 we give an application to theta functions of even, unimodular ...

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