On the vanishing of coefficients of $\eta^{26}

TL;DR

This paper investigates the conditions under which coefficients of the modular form ta^{26} vanish, building on Serre's sufficient condition and exploring the necessity through Hecke eigenforms.

Contribution

It provides partial results towards establishing the necessity of Serre's condition for the vanishing of ta^{26} coefficients using Hecke eigenform theory.

Findings

Partial cases where the converse condition holds

Extension of Serre's results on ta^{26} coefficients

Application of Hecke eigenform theory to modular form coefficient analysis

Abstract

J.-P. Serre, in his paper [1], established a sufficient condition on $n$ for the $n$-th coefficient of the series $\eta^{26}$ to vanish. However, the question that whether this is a necessary condition remained unanswered. In this paper, using the theory of Hecke eigenforms explored by Serre, we prove some partial cases for the converse part.