Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace

2024/07/16 by William Cason, Cason, William, Akash Jim +9 · 2 citations
Mathematics · #11F11 #11F25 #11F72 #Advanced Algebra and Geometry #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2407.11694

openalex publication_date 2024/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For m ≥ 1, let N ≥ 1 be coprime to m, k ≥ 2, and χ be a Dirichlet character modulo N with χ(-1)=(-1)^k. Then let T_m^new(N,k,χ) denote the restriction of the m-th Hecke operator to the space S_k^new(Γ₀(N), χ). We demonstrate that for fixed m and trivial character χ, the second coefficient of the characteristic polynomial of T_m^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^new(Γ₀(N), χ) is trivial, the second coefficient of the characteristic polynomial of T_m^new(N,k,χ) vanishes for only finitely many triples (N,k,χ).

Cited by

• Hecke Eigenvalue Equidistribution over the Newspaces with Nebentypus
• Dimension sequences of modular forms

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• Factoring Hecke polynomials modulo a prime
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