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.

Comments are welcome

References in corpus (1)

• Images of adelic Galois representations for modular forms