Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace

TL;DR

This work analyzes the second coefficient $a_2^{\text{new}}(m,N,k,\chi)$ of the Hecke polynomial on the new subspace $S_k^{\text{new}}(\Gamma_0(N),\chi)$ by expressing it in terms of traces of Hecke operators and applying the Eichler-Selberg trace formula. By bounding the new-trace components $A_i^{\text{new}}$ via Dirichlet-convolution techniques and explicit class-number bounds, the authors establish finiteness results for the nonvanishing of $a_2^{\text{new}}$: for trivial character and fixed $m$, $a_2^{\text{new}}$ is nonzero only for finitely many $(N,k)$, with a precise sign bias depending on whether $m$ is a square. They explicitly compute all vanishing pairs for $m=2$ and $m=4$ with trivial character and provide analogous finite-vanishing results in the general character case, after excluding infinitely many trivial cases. The paper further discusses how these second-coefficient phenomena relate to broader questions about traces, eigenforms, and Maeda-type conjectures, and outlines extensions to higher coefficients and other subspaces. Overall, the results give concrete finite exceptional sets and reveal a robust sign bias in $a_2^{\text{new}}$, highlighting its potential as a diagnostic tool for the structure of newforms.

Abstract

For $m \geq 1$, let $N \geq 1$ be coprime to $m$, $k \geq 2$, and $χ$ be a Dirichlet character modulo $N$ with $χ(-1)=(-1)^k$. Then let $T_m^{\text{new}}(N,k,χ)$ denote the restriction of the $m$-th Hecke operator to the space $S_k^{\text{new}}(Γ_0(N), χ)$. We demonstrate that for fixed $m$ and trivial character $χ$, the second coefficient of the characteristic polynomial of $T_m^{\text{new}}(N,k)$ vanishes for only finitely many pairs $(N,k)$, and we further determine the sign. To demonstrate our method, for $m=2,4$, we also compute all pairs $(N,k)$ for which the second coefficient vanishes. In the general character case, we also show that excluding an infinite family where $S_k^{\text{new}}(Γ_0(N), χ)$ is trivial, the second coefficient of the characteristic polynomial of $T_m^{\text{new}}(N,k,χ)$ vanishes for only finitely many triples $(N,k,χ)$.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2: knightly-li, cohen-stromberg
• Lemma 2.3: cohen-stromberg
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3