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 $k\geq 4$, 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 $[5k+1+\sqrt{\log(k)}]$ distinct prime factors. We give examples of about hundred forms to which our theorem applies.

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