# Computing coefficients of modular forms

# Computing coefficients of modular forms

This chapter applies the main result on the computation of Galois representations attached to modular forms of level one to the computation of coefficients of modular forms. It treats the case of the discriminant modular form, that is, the computation of Ramanujan's tau-function at primes, and then deals with the more general case of forms of level one and arbitrary weight *k*, reformulated as the computation of Hecke operators *T*ⁿ as ℤ-linear combinations of the *T*ᵢ with *i* < *k* = 12. The chapter gives an application to theta functions of even, unimodular positive definite quadratic forms over ℤ.

*Keywords:*
modular forms, Galois representations, coefficients, Ramanujan's tau-function, Hecke operators

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