A variant of Lehmer's conjecture (Q868900)

Let \[ f(z)=\sum_{n=1}^\infty a_ne^{2\pi i nz} \] be the Fourier expansion of a normalized eigenform of weight \(\geq\) 2 and suppose that the \(a_n\) are rational integers for all \(n\). The author proves \[ \#\{n\leq x: (n,a_n)=1\}\ll \frac{x}{\log\log\log x}. \] The implied constant depends on the level of \(f\).