{
  "abstract": "FINDING: Theta series of even unimodular lattices (E8, Leech) are modular forms whose coefficients encode lattice point counts, with divisibility properties tied to the lattice's structure. | MATH: For an even unimodular lattice \\(L\\) of rank \\(n\\), the theta series \\(\\Theta_L(\\tau) = \\sum_{v \\in L} q^{v \\cdot v / 2}\\) (with \\(q = e^{2\\pi i \\tau}\\)) is a modular form of weight \\(n/2\\) for \\(\\mathrm{SL}_2(\\mathbb{Z})\\). For E8 (rank 8), \\(\\Theta_{E8} = 1 + 240 \\sum_{k=1}^\\infty \\sigma_3(k) q^k\\), where \\(\\sigma_3(k)\\) is the sum of cubes of divisors. For the Leech lattice (rank 24), \\(\\Theta_{\\Lambda_{24}} = E_{12}(\\tau) - \\frac{65520}{691} \\Delta(\\tau)\\), where \\(E_{12}\\) is the Eisenstein series of weight 12 and \\(\\Delta\\) is the discriminant (cusp form). The coefficient of \\(q^m\\) in \\(\\Theta_{\\Lambda_{24}}\\) is divisible by 1440 for all \\(m \\ge 1\\) — this follows from the fact that \\(\\Theta_{\\Lambda_{24}} \\equiv 1 \\pmod{1440}\\) as a modular form, a consequence of the lattice being e\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fy55v6d5070a1.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23587142",
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  "id": "fy55v6d5070a1",
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  "landing_url": "https://zenodo.org/records/23587142",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fy55v6d5070a1.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T12:21:52Z",
      "native_id": "23587142",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23587142"
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  "subjects": [],
  "title": "Theta Series of Even Unimodular Lattices as Modular Forms Encoding Point Counts — E8 Intelligence Research",
  "title_text": "Theta Series of Even Unimodular Lattices as Modular Forms Encoding Point Counts — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-06T12:21:52Z",
  "url": "https://research.draconitic.com/paper/fy55v6d5070a1",
  "venue": "Zenodo",
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}