Computing coefficients of modular forms

We consider normalised Hecke eigenforms of level n and weight k.  In
recent years, Edixhoven, Couveignes et al. (for n = 1) and the speaker
(generalisation to n > 1) developed an algorithm that, given such an f
and a positive integer m in factored form, computes the m-th coefficient
of the q-expansion of f, and whose running time is polynomial in n, k
and log m under the generalised Riemann hypothesis.  I will explain this
algorithm and the fundamental idea behind it, namely the computation of
two-dimensional Galois representations over finite fields attached to f.