{
  "id": "1602.05745",
  "version": "v1",
  "published": "2016-02-18T10:27:51.000Z",
  "updated": "2016-02-18T10:27:51.000Z",
  "title": "On the gaps between non-zero Fourier coefficients of cusp forms of higher weight",
  "authors": [
    "Narasimha Kumar"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that if a modular cuspidal eigenform $f$ of weight $2k$ is $2$-adically close to an elliptic curve $E/\\mathbb{Q}$, which has a cyclic rational $4$-isogeny, then $n$-th Fourier coefficient of $f$ is non-zero in the short interval $(X, X + cX^{\\frac{1}{4}})$ for all $X \\gg 0$ and for some $c > 0$. We use this fact to produce non-CM cuspidal eigenforms $f$ of level $N>1$ and weight $k > 2$ such that $i_f(n) \\ll n^{\\frac{1}{4}}$ for all $n \\gg 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-18T10:27:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11F11",
      "11F33",
      "11G05"
    ],
    "keywords": [
      "non-zero fourier coefficients",
      "higher weight",
      "cusp forms",
      "produce non-cm cuspidal eigenforms",
      "modular cuspidal eigenform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160205745K"
    }
  }
}