A5 Artin a_p
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 Persistent Homology:n_significant_features (+3.8z) — beyond what the standard bank predicts for it. It sits beside Categorical Sensor in the atlas (standard-bank rank 57) — a neighbor conventional features miss.
Persistent Homology:n_significant_features | +3.8z | bank-miss 3.0σ |
Composition
Fixed alphabet — only 14 distinct symbols across 16384 samples.
Render Gallery
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Categorical Sensor | 4.06 | cross-domain |
| Collatz Gap Lengths | 4.31 | |
| Poker Hands | 4.38 | cross-domain |
Which Geometries Light Up
Gottwald-Melbourne › Gottwald-Melbourne:angular_spectral_structure | rank 296/298 | 0.4357 |
Julia Set › Julia Set:escape_entropy | rank 4/298 | 4.4545 |
Persistent Homology › Persistent Homology:n_significant_features | rank 2/298 | 25.4000 |