{
  "id": "2008.08768",
  "version": "v1",
  "published": "2020-08-20T04:26:47.000Z",
  "updated": "2020-08-20T04:26:47.000Z",
  "title": "Bias of Root Numbers for Modular Newforms of Cubic Level",
  "authors": [
    "Qinghua Pi",
    "Zhi Qi"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $H^{\\pm}_{2k} (N^3)$ denote the set of modular newforms of cubic level $N^3$, weight $2 k$, and root number $\\pm 1$. For $N > 1$ squarefree and $k>1$, we use an analytic method to establish neat and explicit formulas for the difference $|H^{+}_{2k} (N^3)| - |H^{-}_{2k} (N^3)|$ as a multiple of the product of $\\varphi (N)$ and the class number of $\\mathbb{Q}(\\sqrt{- N})$. In particular, the formulas exhibit a strict bias towards the root number $+1$. Our main tool is a root-number weighted simple Petersson formula for such newforms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-20T04:26:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "11F72"
    ],
    "keywords": [
      "root number",
      "modular newforms",
      "cubic level",
      "root-number weighted simple petersson formula",
      "strict bias"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}