A variant of Lehmer's conjecture, II: The CM-case

Abstract

Let $f$ be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer $n$ has a factor common with the $n$-th Fourier coefficient of $f$. The second author \cite{kumar3} showed that this happens very often. In this paper, we give an asymptotic formula for the number of integers $n$ for which $(n, a(n))=1$, where $a(n)$ is the $n$-th Fourier coefficient of a normalized Hecke eigenform $f$ of weight $2$ with rational integer Fourier coefficients and has complex multiplication.