{"paper":{"title":"On the Hecke Eigenvalues of Maass Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fan Zhou, Wenzhi Luo","submitted_at":"2014-05-20T03:08:43Z","abstract_excerpt":"Let $\\phi$ denote a primitive Hecke-Maass cusp form for $\\Gamma_o(N)$ with the Laplacian eigenvalue $\\lambda_\\phi=1/4+t_{\\phi}^2$. In this work we show that there exists a prime $p$ such that $p\\nmid N$, $|\\alpha_{p}|=|\\beta_{p}| = 1$, and $p\\ll(N(1+|t_{\\phi}|))^c$, where $\\alpha _{p},\\;\\beta _{p}$ are the Satake parameters of $\\phi$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $0.27332$. In addition, we prove that the natural density of such primes $p$ ($p\\nmid N$ and $|\\alpha_{p}|=|\\beta_{p}| = 1$) is at least $34/35$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4937","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}