🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 139 β€Ί 139.4.a β€Ί 139.4.a.b

Newform orbit 139.4.a.b

Trace form

The 12 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

12qβˆ’9q2βˆ’7q3+39q4βˆ’67q5βˆ’6q6βˆ’102q8+33q9βˆ’29q10βˆ’201q11βˆ’143q12+77q13βˆ’197q14βˆ’48q15βˆ’q16βˆ’310q17βˆ’359q18+127q19βˆ’451q20βˆ’496q21βˆ’17q22βˆ’392q23βˆ’360q24+O(q25)12q - 9q^{2} - 7q^{3} + 39q^{4} - 67q^{5} - 6q^{6} - 102q^{8} + 33q^{9} - 29q^{10} - 201q^{11} - 143q^{12} + 77q^{13} - 197q^{14} - 48q^{15} - q^{16} - 310q^{17} - 359q^{18} + 127q^{19} - 451q^{20} - 496q^{21} - 17q^{22} - 392q^{23} - 360q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 12. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x12+16x11βˆ’112x10βˆ’2049x9+5544x8+75971x7βˆ’201041x6βˆ’696191x5+1621237x4+2580938x3βˆ’4359735x2βˆ’3560985x+3054264x^{12} + 16x^{11} - 112x^{10} - 2049x^{9} + 5544x^{8} + 75971x^{7} - 201041x^{6} - 696191x^{5} + 1621237x^{4} + 2580938x^{3} - 4359735x^{2} - 3560985x + 3054264

Atkin–Lehner signs

ppwpw_p
139βˆ’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’9βˆ’7βˆ’670βˆ’20177βˆ’310127βˆ’392βˆ’993βˆ’29444βˆ’689371143βˆ’1903βˆ’939βˆ’673βˆ’477βˆ’2722728βˆ’226βˆ’1773βˆ’2418650

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 0.112 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, 139]: [1, 36] (certified at ell = 2147483137)
  • 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(139, 4)[1]   # 139.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 139:4 1000