Signs of the Second Coefficients of Hecke Polynomials

TL;DR

This paper investigates the behavior of the second coefficients of Hecke polynomials, showing nonvanishing for most cases and determining their signs for various fixed parameters, with explicit computations for specific levels.

Contribution

It establishes the nonvanishing of the second coefficient for almost all triples and determines the sign of the second coefficient in many cases, including explicit computations for certain levels.

Findings

Second coefficient is nonvanishing for all but finitely many triples.

Sign of the second coefficient is determined for almost all pairs when the character is trivial.

Explicit sign computations are provided for levels 3 and 4 with trivial character.

Abstract

Let $T_m(N, k, \chi)$ be the $m$-th Hecke operator of level $N$, weight $k \ge 2$, and nebentypus $\chi$, where $N$ is coprime to $m$. We first show that for any given $m \ge 1$, the second coefficient of the characteristic polynomial of $T_m(N, k, \chi)$ is nonvanishing for all but finitely many triples $(N,k,\chi)$. Furthermore, for $\chi$ trivial and any fixed $m$, we determine the sign of the second coefficient for all but finitely many pairs $(N,k)$. Finally, for $\chi$ trivial and $m=3,4$, we compute the sign of the second coefficient for all pairs $(N,k)$.