🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 137 β€Ί 137.4.a β€Ί 137.4.a.a

Newform orbit 137.4.a.a

Trace form

The 15 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

15qβˆ’6q2βˆ’18q3+44q4βˆ’20q5βˆ’55q6βˆ’122q7βˆ’90q8+97q9βˆ’111q10βˆ’72q11βˆ’184q12βˆ’264q13βˆ’72q14βˆ’160q15+20q16βˆ’146q17βˆ’261q18βˆ’286q19βˆ’305q20βˆ’230q21βˆ’289q22βˆ’380q23βˆ’652q24+O(q25)15q - 6q^{2} - 18q^{3} + 44q^{4} - 20q^{5} - 55q^{6} - 122q^{7} - 90q^{8} + 97q^{9} - 111q^{10} - 72q^{11} - 184q^{12} - 264q^{13} - 72q^{14} - 160q^{15} + 20q^{16} - 146q^{17} - 261q^{18} - 286q^{19} - 305q^{20} - 230q^{21} - 289q^{22} - 380q^{23} - 652q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 15. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x15+24x14+10x13βˆ’2945x12βˆ’10142x11+116511x10+383221x9βˆ’2352344x8βˆ’4675588x7+25331967x6+11100081x5βˆ’114567032x4+85099013x3+29352834x2βˆ’10605996xβˆ’2664063x^{15} + 24x^{14} + 10x^{13} - 2945x^{12} - 10142x^{11} + 116511x^{10} + 383221x^{9} - 2352344x^{8} - 4675588x^{7} + 25331967x^{6} + 11100081x^{5} - 114567032x^{4} + 85099013x^{3} + 29352834x^{2} - 10605996x - 2664063

Atkin–Lehner signs

ppwpw_p
137βˆ’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’6βˆ’18βˆ’20βˆ’122βˆ’72βˆ’264βˆ’146βˆ’286βˆ’380βˆ’4βˆ’874βˆ’160640βˆ’486βˆ’148454246βˆ’380βˆ’1624βˆ’494βˆ’1260βˆ’2856βˆ’2252βˆ’314βˆ’3840

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 0.111 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, 137]: [1, 36] (certified at ell = 2147483249)
  • T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (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(137, 4)[0]   # 137.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 137:4 1000