Divisors of Fourier coefficients of two newforms

TL;DR

This paper estimates the distribution of primes related to the divisors of Fourier coefficients of two distinct non-CM newforms, providing bounds on primes with specific congruence properties under GRH.

Contribution

It introduces new bounds on primes associated with Fourier coefficient differences of two non-CM newforms, including applications to congruences and multiplicity one results.

Findings

Bound on the number of primes with small divisor count of Fourier coefficient differences

Bound on primes producing congruences between two newforms

Establishment of a multiplicity one result via congruences

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.