{
  "id": "1611.06967",
  "version": "v1",
  "published": "2016-11-21T19:37:22.000Z",
  "updated": "2016-11-21T19:37:22.000Z",
  "title": "Newforms with rational coefficients",
  "authors": [
    "David P. Roberts"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider the set of classical newforms with rational coefficients and no complex multiplication. We study the distribution of quadratic-twist classes of these forms with respect to weight $k$ and minimal level $N$. We conjecture that for each weight $k \\geq 6$, there are only finitely many classes. In large weights, we make this conjecture effective: in weights $18 \\leq k \\leq 24$, all classes have $N \\leq 30$, in weights $26 \\leq k \\leq 50$, all classes have $N \\in \\{2,6\\}$, and in weights $k \\geq 52$, there are no classes at all. We study some of the newforms appearing on our conjecturally complete list in more detail, especially in the cases $N=2$, $3$, $4$, $6$, and $8$, where formulas can be kept nearly as simple as those for the classical case $N=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-21T19:37:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11"
    ],
    "keywords": [
      "rational coefficients",
      "minimal level",
      "complex multiplication",
      "conjecturally complete list",
      "large weights"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}