🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 127 β€Ί 127.4.a β€Ί 127.4.a.b

Newform orbit 127.4.a.b

Trace form

The 13 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

13qβˆ’8q2βˆ’16q3+34q4βˆ’46q5βˆ’2q6βˆ’26q7βˆ’117q8+3q9βˆ’25q10βˆ’53q11βˆ’255q12βˆ’75q13βˆ’152q14βˆ’62q15+82q16βˆ’479q17βˆ’292q18βˆ’209q19βˆ’533q20βˆ’174q21βˆ’407q22βˆ’376q23+108q24+O(q25)13q - 8q^{2} - 16q^{3} + 34q^{4} - 46q^{5} - 2q^{6} - 26q^{7} - 117q^{8} + 3q^{9} - 25q^{10} - 53q^{11} - 255q^{12} - 75q^{13} - 152q^{14} - 62q^{15} + 82q^{16} - 479q^{17} - 292q^{18} - 209q^{19} - 533q^{20} - 174q^{21} - 407q^{22} - 376q^{23} + 108q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 13. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x13+24x12+44x11βˆ’2104x10βˆ’8153x9+58690x8+267426x7βˆ’567203x6βˆ’2973938x5+1589288x4+11202964x3+15981x2βˆ’8708440xβˆ’34772x^{13} + 24x^{12} + 44x^{11} - 2104x^{10} - 8153x^{9} + 58690x^{8} + 267426x^{7} - 567203x^{6} - 2973938x^{5} + 1589288x^{4} + 11202964x^{3} + 15981x^{2} - 8708440x - 34772

Atkin–Lehner signs

ppwpw_p
127βˆ’1-1

For pβˆ₯Np \parallel N every form in the orbit has ap=βˆ’pk/2βˆ’1wpa_p = -p^{k/2-1} w_p. The Fricke sign is wN=βˆ’1w_N = -1, so the sign of the functional equation is (βˆ’1)k/2wN=βˆ’1(-1)^{k/2} w_N = -1: each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

pp2357111317192329313741434753596167717379838997
tr⁑ap\operatorname{tr} a_pβˆ’8βˆ’16βˆ’46βˆ’26βˆ’53βˆ’75βˆ’479βˆ’209βˆ’376βˆ’158βˆ’307βˆ’171βˆ’641530βˆ’555βˆ’1640βˆ’860191912115βˆ’2563169βˆ’1688βˆ’2752βˆ’2124

tr⁑an\operatorname{tr} a_n for every n≀1000n \le 1000 is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of TT over Z\mathbb{Z} is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks
  • dim M^+ at levels [1, 127]: [1, 33] (certified at ell = 2147483647)
  • T = [(2, 1), (3, 1)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 5 primes, 0 rejected
Open in the notebook Β· JSON Β· stored data
Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(127, 4)[1]   # 127.4.a.b

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 127:4 1000