{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/nonvanishing-of-second-coefficients-of-hecke","title":"Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace","arxiv_id":"2407.11694","date":"2024-07-16","proceeding":null,"authors":["William Cason","Akash Jim","Charlie Medlock","Erick Ross","Trevor Vilardi","Hui Xue"],"abstract":"For $m \\geq 1$, let $N \\geq 1$ be coprime to $m$, $k \\geq 2$, and $\\chi$ be a Dirichlet character modulo $N$ with $\\chi(-1)=(-1)^k$. Then let $T_m^{\\text{new}}(N,k,\\chi)$ denote the restriction of the $m$-th Hecke operator to the space $S_k^{\\text{new}}(\\Gamma_0(N), \\chi)$. We demonstrate that for fixed $m$ and trivial character $\\chi$, 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 $S_k^{\\text{new}}(\\Gamma_0(N), \\chi)$ is trivial, the second coefficient of the characteristic polynomial of $T_m^{\\text{new}}(N,k,\\chi)$ vanishes for only finitely many triples $(N,k,\\chi)$.","url_abs":"https://arxiv.org/abs/2407.11694v2","url_pdf":"https://arxiv.org/pdf/2407.11694v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"nonvanishing-of-second-coefficients-of-hecke","repo_url":"https://github.com/eross156/nonvanishing-a2-on-new-subspace","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}