Remarks on the Fourier coefficients of modular forms

AbstractWe 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 Γ0(M) with trivial Nebentypus χ0 and with integer Fourier coefficients, let Np(f)=χ0(p)pk−1+1−ap(f) (here ap(f) is the p-th-Fourier coefficient of f). We show under GRH and Artinʼs Holomorphy Conjecture that there are infinitely many p such that Np(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 Np(f) is of normal order log(log(p)) and that the distribution of these values is asymptotically a Gaussian distribution (“Erdős–Kac type theorem”)