{"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/variants-of-lehmer-s-speculation-for-newforms","title":"Variants of Lehmer's speculation for newforms","arxiv_id":"2005.10354","date":"2020-05-20","proceeding":null,"authors":["Jennifer S. Balakrishnan","William Craig","Ken Ono","Wei-Lun Tsai"],"abstract":"In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of $\\tau(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform $f(z)$. We offer a method, which applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $3\\leq \\ell\\leq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $\\tau(n)$ with $n>1$, we prove that $$\\tau(n)\\not \\in \\{\\pm 1, \\pm 3, \\pm 5, \\pm 7, \\pm 13, \\pm 17, -19, \\pm 23, \\pm 37, \\pm 691\\},$$ and assuming GRH we show for primes $\\ell$ that $$\\tau(n)\\not \\in \\left \\{ \\pm \\ell\\ : \\ 41\\leq \\ell\\leq 97 \\ {\\textrm{with}}\\ \\left(\\frac{\\ell}{5}\\right)=-1\\right\\} \\cup \\left \\{ -11, -29, -31, -41, -59, -61, -71, -79, -89\\right\\}. $$ We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $\\ell$, we prove that $\\pm \\ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{\\pm}(\\ell,m)=O_{\\ell}(m)$ and even level coprime to $\\ell,$ where $M^{\\pm}(\\ell,m)$ is effectively computable.","url_abs":"https://arxiv.org/abs/2005.10354v4","url_pdf":"https://arxiv.org/pdf/2005.10354v4.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":"variants-of-lehmer-s-speculation-for-newforms","repo_url":"https://github.com/jbalakrishnan/Lehmer","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}