Coprimality of Fourier coefficients of eigenforms
Volume 203 / 2022
Acta Arithmetica 203 (2022), 69-96
MSC: Primary 11F30; Secondary 11F80, 11N37, 11N64, 11N99.
DOI: 10.4064/aa210817-8-2
Published online: 24 March 2022
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 $(a_1(p), a_2(p))$.
