Coprimality of Fourier coefficients of eigenforms

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 $(a_1(p),a_2(p))$.