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 Np(E)=#E(Fp)=p+1-ap(E) is also a prime. We consider a variant of this question. For a newform f, without CM, of weight k≥4, on Γ ₀(M) with trivial Nebentypus χ ₀ and with integer Fourier coefficients, let N _p(f)=χ 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+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 ("Erdo{double acute}s-Kac type theorem").