A5 Artin a_p

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number-theoryarithmetic

What It Is

Hecke traces a_p for 100 weight-1 cuspidal newforms with projective Galois image A_5 (icosahedral), pooled across LMFDB labels for primes p ≤ 1999 (303 a_p per form, 30,300 pooled). Each a_p is the integer trace of Frobenius in a 2-dimensional Artin representation; the icosahedral case is the Buhler 1978 modularity result, completed for all 2-dim odd Artin reps by Khare–Wintenberger / Kisin. Values lie in the finite set {+32, ±16, ±12, ±8, ±6, ±4, ±2, 0} (all even, since each is the trace down to ℤ of an even-degree coefficient field); the pooled distribution follows the Chebotarev density theorem over A_5 conjugacy classes, heavily zero-biased (~64%) and symmetric. Distinct universality class from continuous Sato-Tate semicircle: a discrete, finite-support Galois-distributed integer stream — the standard 'arithmetic trace' calibration anchor for metrics responding to discrete-categorical structure with deep number-theoretic provenance.

Interpretation

Standard analysis sees: no strongly notable standard properties. The atlas finds no named structure, but the source is distinctively extreme on Level Statistics:spacing_gue_distance (+2.8z) — beyond what the standard bank predicts for it. It sits beside Categorical Sensor in the atlas (standard-bank rank 60) — a neighbor conventional features miss.

What the atlas adds

Atlas-extreme metrics the standard bank can’t predict for this source

Level Statistics:spacing_gue_distance +2.8z bank-miss 1.3σ