  We propose an improved algorithm for computing mod $\ell$ Galois
representations associated to a cusp form $f$ of level one. The proposed method
allows us to explicitly compute the case with $\ell=29$ and $f$ of weight
$k=16$, and the cases with $\ell=31$ 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.
