{
  "id": "1405.4937",
  "version": "v2",
  "published": "2014-05-20T03:08:43.000Z",
  "updated": "2014-06-18T16:10:34.000Z",
  "title": "On the Hecke Eigenvalues of Maass Forms",
  "authors": [
    "Wenzhi Luo",
    "Fan Zhou"
  ],
  "comment": "Version 2: typos corrected and a new section on natural density added",
  "categories": [
    "math.NT"
  ],
  "abstract": "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 $34/35$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-18T16:10:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11F41",
      "11F12"
    ],
    "keywords": [
      "maass forms",
      "hecke eigenvalues",
      "primitive hecke-maass cusp form",
      "satake parameters",
      "natural density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.4937L"
    }
  }
}