On the coefficients of an eigenform

•We establish an estimate on the coefficients of the eigenforms.•Our result is related to the well known Lehmerʼs conjecture.•The combinatorial sieve is used in our proof.

Lehmerʼs conjecture on Ramanujanʼs tau function τ(n) asserts that τ(n)≠0 for any n⩾1. We consider a variant of Lehmerʼs conjecture. Let f(z)=∑n=1∞ane2πinz be a normalized eigenform, where the an are rational integers for all n, and let g(n) be a polynomial with integer coefficients. We bound the number of n⩽x such that an and g(n) have no common factor.