We propose an improved algorithm for computing mod β Galois
representations associated to a cusp form f of level one. The proposed method
allows us to explicitly compute the case with β=29 and f of weight
k=16, and the cases with β=31 and f of weight k=12,20,22. All the
results are rigorously proved to be correct.
As an example, we will compute the values modulo 31 of Ramanujan's tau
function at some huge primes up to a sign. Also we will give an improved higher
bound on Lehmer's conjecture for Ramanujan's tau function.Comment: This paper has been published in Acta Arithmetic