{"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/signs-of-the-second-coefficients-of-hecke","title":"Signs of the Second Coefficients of Hecke Polynomials","arxiv_id":"2407.10951","date":"2024-07-15","proceeding":null,"authors":["Erick Ross","Hui Xue"],"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 $(N,k)$.","url_abs":"https://arxiv.org/abs/2407.10951v1","url_pdf":"https://arxiv.org/pdf/2407.10951v1.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":"signs-of-the-second-coefficients-of-hecke","repo_url":"https://github.com/eross156/nonvanishing-a2-coefficient","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}