Remarks on the Fourier coefficients of modular forms

We consider a variant of a question of N. Koblitz. For an elliptic curve E/Q which is not Q-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes p such that N-p(E) = #E(F-p) = p + 1 - a(p)(E) is also a prime. We consider a variant of this question. For a newform f, without CM, of weight k >= 4, on Gamma(0)(M) with trivial Nebentypus chi(0) and with integer Fourier coefficients, let N-p(f) = chi(0)(p)p(k-1) + 1 -a(p)(f) (here a(p)(f) is the p-th-Fourier coefficient of f). We show under GRH and Artin's Holomorphy Conjecture that there are infinitely many p such that N-p(f) has at most [5k + 1 + root log(k)] distinct prime factors. We give examples of about hundred forms to which our theorem applies. We also show, on GRH. that the number of distinct prime factors of N-p(f) is of normal order log(log(p)) and that the distribution of these values is asymptotically a Gaussian distribution ("Erdos-Kac type theorem"). (C) 2012 Elsevier Inc. All rights reserved.