{
  "id": "1008.4008",
  "version": "v1",
  "published": "2010-08-24T10:53:51.000Z",
  "updated": "2010-08-24T10:53:51.000Z",
  "title": "A new basis for the space of modular forms",
  "authors": [
    "Shinji Fukuhara"
  ],
  "comment": "AMS-LaTeX, 6 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $G_{2n}$ be the Eisenstein series of weight $2n$ for the full modular group $\\Gamma=SL_2(\\ZZ)$. It is well-known that the space $M_{2k}$ of modular forms of weight $2k$ on $\\Gamma$ has a basis $\\{G_{4}^\\alpha G_{6}^\\beta\\ |\\ \\alpha,\\beta\\in\\ZZ,\\ \\alpha,\\beta\\geq 0,\\ 4\\alpha+6\\beta=2k\\}$. In this paper we will exhibit another (simpler) basis for $M_{2k}$. It is given by $\\{G_{2k}\\}\\cup\\{G_{4i}G_{2k-4i}\\ |\\ i=1,2,\\ldots,d_k\\}$ if $2k\\equiv 0\\pmod 4$, and $\\{G_{2k}\\}\\cup\\{G_{4i+2}G_{2k-4i-2}\\ |\\ i=1,2,\\ldots,d_k\\}$ if $2k\\equiv 2\\pmod 4$ where $d_k+1=\\dim_{\\CC} M_{2k}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-08-24T10:53:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "11F67",
      "11F30"
    ],
    "keywords": [
      "modular forms",
      "full modular group",
      "eisenstein series",
      "well-known"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.4008F"
    }
  }
}