Divisors of Fourier coefficients of two newforms

Arvind Kumar and Moni Kumari

Vol. 326 (2023), No. 1, 85–107

DOI: 10.2140/pjm.2023.326.85

Abstract

For a pair of distinct non-CM newforms of weights at least 2 and 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

$$\omega (a_1(p)-a_2(p))\le \left[7k+\frac{1}{2}+k^{1∕5}\right],$$

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 $\left[7k+\frac{1}{2}+k^{1∕5}\right]$. We also obtain a multiplicity-one result for newforms via congruences.

Keywords

modular forms, Fourier coefficients, Galois representations, Richert sieve

Mathematical Subject Classification

Primary: 11F30, 11N36

Secondary: 11F33, 11F80

Milestones

Received: 11 April 2022

Revised: 27 July 2023

Accepted: 18 October 2023

Published: 14 December 2023