{
  "id": "2407.08881",
  "version": "v1",
  "published": "2024-07-11T21:56:57.000Z",
  "updated": "2024-07-11T21:56:57.000Z",
  "title": "Newspaces with Nebentypus: An Explicit Dimension Formula, Classification of Trivial Newspaces, and Character Equidistribution Property",
  "authors": [
    "Erick Ross"
  ],
  "comment": "36 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Consider $N \\geq 1$, $k \\geq 2$, and $\\chi$ a Dirichlet character modulo $N$ such that $\\chi(-1) = (-1)^k$. For any bound $B$, one can show that $\\dim S_k(\\Gamma_0(N),\\chi) \\le B$ for only finitely many triples $(N,k,\\chi)$. It turns out that this property does not extend to the newspace; there exists an infinite family of triples $(N,k,\\chi)$ for which $\\dim S_k^{\\text{new}}(\\Gamma_0(N),\\chi) = 0$. However, we classify this case entirely. We also show that excluding the infinite family for which $\\dim S_k^{\\text{new}}(\\Gamma_0(N),\\chi) = 0$, $\\dim S_k^{\\text{new}}(\\Gamma_0(N),\\chi) \\leq B$ for only finitely many triples $(N,k,\\chi)$. In order to show these results, we derive an explicit dimension formula for the newspace $S_k^{\\text{new}}(\\Gamma_0(N),\\chi)$. We also use this explicit dimension formula to prove a character equidistribution property and disprove a conjecture from Greg Martin that $\\dim S_2^{\\text{new}}(\\Gamma_0(N))$ takes on all possible non-negative integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-11T21:56:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "11F06"
    ],
    "keywords": [
      "explicit dimension formula",
      "character equidistribution property",
      "trivial newspaces",
      "classification",
      "nebentypus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}