Approximate Atkin-Serre Conjecture

Abstract

Let $\lambda(m)$ be the $m$th coefficient of a modular form $f(z)=\sum_{m\geq 1} \lambda(m)q^m$ of weight $k\geq 4$, let $p^n$ be a prime power, and let $\varepsilon>0$ be a small number. An approximate of the Atkin-Serre conjecture on the lower bound of the form $\left |\lambda\left (p^n\right )\right | \geq p^{(k-1)n/2-2k+2\varepsilon}$ is presented in this note.