Atkin–Serre conjecture for Fourier coefficients at composite indices

Let $f$ be a newform of weight $k \geq 4$ and level $N$, with Fourier coefficients $a_f(m)$. For every $\varepsilon>0$, there exists a constant $c_\varepsilon>0$ such that for every $m\geq 1$, if $a_f(m)\neq 0$, then

$$\left\lvert a_f(m) \right\rvert \geq c_\varepsilon m^{(k-3)/2-\varepsilon}.$$

Generalized Atkin–Serre conjecture. Under these hypotheses, the stated lower bound holds for every nonzero Fourier coefficient $a_f(m)$. This extends the Atkin–Serre conjecture from prime-indexed to general Fourier coefficients; the source presents it as a proposed conjecture, with no resolution supplied.