We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa164-4-5,
author = {Peng Tian},
title = {Computations of Galois representations associated to modular forms of level one},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {399-411},
zbl = {1334.11092},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa164-4-5}
}
Peng Tian. Computations of Galois representations associated to modular forms of level one. Acta Arithmetica, Tome 166 (2014) pp. 399-411. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa164-4-5/