Nonvanishing of Fourier coefficients of modular forms

Let $f=\sum_{n=1}^\infty a_f(n)q^n$ be a cusp form with integer weight $k\ge 2$that is not a linear combination of forms with complex multiplication. For $n\ge 1$, let

Improving on work of Balog, Ono, and Serre we show that $i_f(n)\ll _{f,\phi}\phi(n)$ for almost all $n$, where $\phi(x)$ is any good function (e.g. such as $\log\log(x)$) monotonically tending to infinity with $x$. Using a result of Fouvry and Iwaniec, if $f$ is a weight 2 cusp form for an elliptic curve without complex multiplication, then we show for all $n$ that $i_f(n)\ll _{f,\varepsilon} n^{\frac{69}{169}+\varepsilon}$. We also obtain conditional results depending on the Generalized Riemann Hypothesis and the Lang-Trotter Conjecture.