Non-repetition of second coefficients of Hecke polynomials

TL;DR

This paper investigates the uniqueness of the second coefficient in the characteristic polynomials of Hecke operators across various parameters, and applies these findings to distinguish Hecke eigenforms.

Contribution

It provides new non-repetition results for the second coefficients of Hecke polynomials in multiple aspects, extending previous work on eigenform distinction.

Findings

Non-repetition results in the horizontal aspect (varying m)

Non-repetition results in the vertical aspect (varying k)

Non-repetition results in the level aspect (varying N)

Abstract

Let $T_m(N,2k)$ denote the $m$-th Hecke operator on the space $S_{2k}(\Gamma_0(N))$ of cuspidal modular forms of weight $2k$ and level $N$. In this paper, we study the non-repetition of the second coefficient of the characteristic polynomial of $T_m(N,2k)$. We obtain results in the horizontal aspect (where $m$ varies), the vertical aspect (where $k$ varies), and the level aspect (where $N$ varies). Finally, we use these non-repetition results to extend a result of Vilardi and Xue on distinguishing Hecke eigenforms.