Computations of Galois representations associated to modular forms of level one
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 uper bound on Lehmer's conjecture for Ramanujan's tau function.