Clifford Rigidity of Transverse Combinations for Weight-4 Newforms

Abstract

We prove that the Fourier coefficients of the three weight-4 newforms of levels 6, 8, 12 are governed by a rigid algebraic framework centred on the even subalgebra Cl⁰(2,2) of the real Clifford algebra of signature (2,2). The coefficients are assembled into a single Clifford element A(n) = a(n)X + b(n)IJ + c(n)ω, and a canonical trace pairing with the ray direction D̃ = X + IJ + ω extracts the invariant s(n) = a(n) − b(n) + c(n), whose rescaled form T(n) = 4s(n) (mod 32) takes values in a discrete octad, satisfies a multiplicative product formula modulo 8, and exhibits a sharp even-index truncation. We prove that Cl(2,2) is the unique Clifford algebra compatible with these constraints, and that the correlator E(m,n) = ⟨v_m, v_n⟩ − s(m)s(n) satisfies the sharp congruence E(m,n) ≡ 0 (mod 384). A systematic scan of all integer-coefficient weight-4 newforms up to level 100 yields 538 mod-8 candidates, of which exactly 91 satisfy the full Clifford even-index conditions; the surviving triples have their IJ-axis of level 6 or 30 and their X- and ω-axes of level divisible by 4. In particular the family contains genuinely new triples beyond the oldform lifts of the original triple (8, 6, 12).