{
  "id": "2302.02725",
  "version": "v1",
  "published": "2023-02-06T12:07:33.000Z",
  "updated": "2023-02-06T12:07:33.000Z",
  "title": "Congruence classes for modular forms over small sets",
  "authors": [
    "Subham Bhakta",
    "S. Krishnamoorthy",
    "R. Muneeswaran"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "J.P. Serre showed that for any integer $m,~a(n)\\equiv 0 \\pmod m$ for almost all $n,$ where $a(n)$ is the $n^{\\text{th}}$ Fourier coefficient of any modular form with rational coefficients. In this article, we consider a certain class of cuspforms and study $\\#\\{a(n) \\pmod m\\}_{n\\leq x}$ over the set of integers with $O(1)$ many prime factors. Moreover, we show that any residue class $a\\in \\mathbb{Z}/m\\mathbb{Z}$ can be written as the sum of at most thirteen Fourier coefficients, which are polynomially bounded as a function of $m.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-06T12:07:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11L07",
      "11P05",
      "11B37",
      "11F80"
    ],
    "keywords": [
      "modular form",
      "small sets",
      "congruence classes",
      "rational coefficients",
      "prime factors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}