Computations Of Fourier Coefficients Of Modular Forms

For large numbers n, it is difficult to compute Ramanujan’s tau function τ(n) defined byRecently, S. J. Edixhoven, J.-M. Couveignes, R. S. de Jong and F. Merkl generalized Schoof’s algorithm and showed a deterministic polynomial time algorithm to compute τ(n). In fact, Edixhoven and Couveignes give a polynomial time algorithm to compute the modular Galois representation and thus as well as the value modulo l of Ramanujan tau function at p. Then combining with the property and the Chinese remainder theorem, one can compute τ(p). It is well known that the repre-sentation appears in the group of l-torsion points of the Jacobian variety of some modular curve X. In fact, their algorithm works for all the modular forms f of level one and boils down to compute a polynomial Pf,l(x) for prime l. Unfortunately the algorithm described by them is difficult to implement. J. Bosman used this algorithm to approximately evaluate Pf,l(x) of modulo l Galois representations associated to modular forms f of level1and of weight k≤22, with l≤23. But since the required precision in the calculations grows quite rapidly with l, Bosman did not compute more cases.In this paper we present an improvement in case gcd(k-2,l+1)>2. In these cases there is a modular curve Xг with Г1(l)≤Г≤Г0(l) with the property that the2-dimensional Galois representation is a subrepresentation of the l-torsion points of the Jacobian of Xг, Therefore we can do the computations with the Jacobian of Xr rather than the Jacobian of X1(l) that Bosman used. Since the genus of Xг is smaller than that of Xl(l), the required precision is smaller and the computation is more efficient. This allows us to deal with cases that were inaccessible by Bosman’s original algorithm.As an example, we succeed to compute the mod31Galois representation associated to discriminant modular form A. For l=29and31, we also compute the mod l Galois representation associated to the unique normalised cusp forms of level1and weights16,20and22. The correctness of each Pf,l is then verified by an application of Serre’s conjecture, proved by Khare-Wintenberger [26]. We compute the values modulo31of Ramanujan’s τ function at some huge primes up to a sign. As a consequence we can verify Lehmer’s conjecture up to a large bound. More precisely, we show that τ(n)≠0, for all n<982149821766199295999Furthermore, we generalize the Lehmer conjecture for level one modular forms. By the results of Swinnerton-Dyer about the congruences of level one modular forms, we explicitly calculate the corresponding bounds.