{
  "abstract": "FINDING: Hecke operators act on spaces of modular forms with a commuting algebra structure whose simultaneous eigenvectors (Hecke eigenforms) encode arithmetic data via their q-expansion coefficients. | MATH: For f = Σ aₙ qⁿ (q = e^{2πiτ}), the Hecke operator Tₚ acts by (Tₚf)(τ) = Σ a_{np} qⁿ + p^{k-1} Σ aₙ q^{np} for weight k. Eigenforms satisfy Tₚf = λₚ f with λₚ = aₚ (normalized, a₁=1). Multiplicativity: a_{mn} = a_m a_n for gcd(m,n)=1; a_{p^{r+1}} = a_p a_{p^r} − p^{k-1} a_{p^{r-1}}. L-function: L(f,s) = Σ aₙ n^{−s} = Π_p (1 − aₚ p^{−s} + p^{k−1−2s})^{−1}. | CONNECTION: The functional equation of L(f,s) relates s ↔ k−s, with critical line Re(s) = k/2. For weight k=2, this gives Re(s)=1, matching the Riemann zeta critical line. The Satake parameters αₚ, βₚ satisfy αₚ + βₚ = aₚ and αₚβₚ = p^{k−1}, giving a 2×2 root-system-like structure per prime. The Ramanujan bound |aₚ| ≤ 2p^{(k−1)/2} is the spectral gap condition, equivalent to αₚ, βₚ lying on a circle of radius p^{(k−1)/2} — a cr\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/bd5pt4vszqzna.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23658560",
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  "id": "bd5pt4vszqzna",
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  "landing_url": "https://zenodo.org/records/23658560",
  "language": "eng",
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  "licence_label": "CC BY 4.0",
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  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd5pt4vszqzna.ris",
  "sources": [
    {
      "first_seen": "2026-11-09T12:21:35Z",
      "native_id": "23658560",
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      "url": "https://zenodo.org/records/23658560"
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  "subjects": [],
  "title": "Hecke Eigenforms Encode Arithmetic Data via q-Expansion Coefficients — E8 Intelligence Research",
  "title_text": "Hecke Eigenforms Encode Arithmetic Data via q-Expansion Coefficients — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:35Z",
  "url": "https://research.draconitic.com/paper/bd5pt4vszqzna",
  "venue": "Zenodo",
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