Coprimality of Fourier coefficients of eigenforms

TL;DR

This paper investigates the coprimality of Fourier coefficients of distinct non-CM eigenforms, providing counts, conjectures on prime densities, and analyzing the average number of prime divisors of their coefficient pairs.

Contribution

It introduces new results on the distribution and coprimality of Fourier coefficients of eigenforms, including conjectures and average order analyses.

Findings

Count of integers with coprime Fourier coefficients

Conjecture on prime density where coefficients are coprime

Analysis of average prime divisors of coefficient pairs

Abstract

Given a pair of distinct non-CM normalized eigenforms having integer Fourier coefficients $a_1 (n)$ and $a_2(n)$, we count positive integers $n$ with $(a_1(n), a_2(n))=1$ and make a conjecture about the density of the set of primes $p$ for which $(a_1(p), a_2(p))=1$. We also study the average order of the number of prime divisors of $(a_1(p), a_2(p))$.