{
  "id": "1608.04196",
  "version": "v1",
  "published": "2016-08-15T07:23:08.000Z",
  "updated": "2016-08-15T07:23:08.000Z",
  "title": "On the gaps between non-zero Fourier coefficients of eigenforms with CM",
  "authors": [
    "Surjeet Kaushik",
    "Narasimha Kumar"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Suppose $E$ is an elliptic curve over $\\mathbb{Q}$ of conductor $N$ with complex multiplication (CM) by $\\mathbb{Q}(i)$, and $f_E$ is the corresponding cuspidal Hecke eigenform in $S^{\\mathrm{new}}_2(\\Gamma_0(N))$. Then $n$-th Fourier coefficient of $f_E$ is non-zero in the short interval $(X, X + cX^{\\frac{1}{4}})$ for all $X \\gg 0$ and for some $c > 0$. As a consequence, we produce infinitely many cuspidal CM eigenforms $f$ level $N>1$ and weight $k > 2$ for which $i_f(n) \\ll n^{\\frac{1}{4}}$ holds, for all $n \\gg 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-15T07:23:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11F11",
      "11F33",
      "11G05"
    ],
    "keywords": [
      "non-zero fourier coefficients",
      "th fourier coefficient",
      "cuspidal cm eigenforms",
      "corresponding cuspidal hecke eigenform",
      "elliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}