Divisors of Fourier coefficients of two newforms

Abstract

For a pair of distinct non-CM newforms of weights at least 2, having rational integral Fourier coefficients $a_{1}(n)$ and $a_{2}(n)$, under GRH, we obtain an estimate for the set of primes $p$ such that $$ \omega(a_1(p)-a_2(p)) \le [ 7k+{1}/{2}+k^{1/5}],$$ where $\omega(n)$ denotes the number of distinct prime divisors of an integer $n$ and $k$ is the maximum of their weights. As an application, under GRH, we show that the number of primes giving congruences between two such newforms is bounded by $[7k+{1}/{2}+k^{1/5} ]$. We also obtain a multiplicity one result for newforms via congruences.