Hecke Eigenforms Encode Arithmetic Data via q-Expansion Coefficients — E8 Intelligence Research

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 = e2πiτ), the Hecke operator Tₚ acts by (Tₚf)(τ) = Σ aₙₚ qⁿ + pᵏ⁻¹ Σ aₙ qⁿᵖ for weight k. Eigenforms satisfy Tₚf = λₚ f with λₚ = aₚ (normalized, a₁=1). Multiplicativity: aₘₙ = a_m a_n for gcd(m,n)=1; a_{pʳ⁺¹} = a_p ap^r − pᵏ⁻¹ a_{pʳ⁻¹}. L-function: L(f,s) = Σ aₙ n⁻ˢ = Π_p (1 − aₚ p⁻ˢ + pᵏ⁻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 = pᵏ⁻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 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com