Twisted moments of the Legendre family, rigid Calabi–Yau modular forms, and a weight-five motive of conductor 1296

Let a(λ) be the Frobenius trace of the Legendre curve y² = x(x−1)(x−λ) over F_p. We study twisted moments Σ ψ(λ) a(λ)^k for characters ψ of order 2, 3, 4, 6. (i) For the quadratic fourth moment we record a closed formula in terms of the newforms 8.4.a.a, 4.6.a.a and 8.6.a.a. It is a direct corollary of trace formulas of Ahlgren and Ahlgren–Ono. (ii) For ψ of order n = 3, 4, 6 we show Re Σ ψ(λ) a(λ)² = −a_f(p) − p − 1, where f = 36.4.a.a, 16.4.a.a, 72.4.a.b are the forms attached by Long–Tu–Yui–Zudilin to rigid hypergeometric Calabi–Yau data. The proof uses Livné's criterion; some standard étale-cohomological inputs are sketched. The identities are verified by exact computation for p ≤ 1000, and for p ≤ 4999 when n = 3. (iii) For the cubic fourth moment we isolate a four-dimensional piece V of weight 5 and Hodge type {0,0,5,5}. Data for all p ≤ 4999 indicate that V is irreducible, not of GL2-type, with Sato–Tate group USp(4). (iv) Descending V to Q gives a degree-four L-function. Numerically, it satisfies a functional equation with conductor 1296 = 2⁴3⁴, sign +1 and gamma factor Γ_C(s)², to precision 1e−15. Its central value is L(W,3) ≈ 2.2413552. It is not among the L-functions currently listed in the LMFDB. We conjecture that it is the spin L-function of a non-cohomological paramodular Siegel cusp form of level 1296 and weight Sym⁴⊗det². All statements are labelled as proved, sketched, or numerical. Code (Python/NumPy/SciPy) and data are included. AI assistance (ChatGPT, Claude) is disclosed in the preprint.