🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 181 β€Ί 181.4.a β€Ί 181.4.a.b

Newform orbit 181.4.a.b

Trace form

The 19 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

19qβˆ’8q2βˆ’24q3+64q4βˆ’16q5βˆ’32q6βˆ’73q7βˆ’150q8+37q9βˆ’105q10βˆ’244q11βˆ’155q12βˆ’96q13βˆ’103q14+56q15+116q16βˆ’30q17βˆ’250q18βˆ’288q19βˆ’396q20βˆ’416q21βˆ’207q22βˆ’648q23βˆ’775q24+O(q25)19q - 8q^{2} - 24q^{3} + 64q^{4} - 16q^{5} - 32q^{6} - 73q^{7} - 150q^{8} + 37q^{9} - 105q^{10} - 244q^{11} - 155q^{12} - 96q^{13} - 103q^{14} + 56q^{15} + 116q^{16} - 30q^{17} - 250q^{18} - 288q^{19} - 396q^{20} - 416q^{21} - 207q^{22} - 648q^{23} - 775q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 19. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x19+32x18+161x17βˆ’4166x16βˆ’41787x15+146202x14+2585516x13βˆ’145916x12βˆ’65493470x11βˆ’53672859x10+749797939x9+536967679x8βˆ’3846317481x7βˆ’87053082x6+7754508502x5βˆ’6787159709x4+1705354273x3+94640317x2βˆ’52200870xβˆ’2944934x^{19} + 32x^{18} + 161x^{17} - 4166x^{16} - 41787x^{15} + 146202x^{14} + 2585516x^{13} - 145916x^{12} - 65493470x^{11} - 53672859x^{10} + 749797939x^{9} + 536967679x^{8} - 3846317481x^{7} - 87053082x^{6} + 7754508502x^{5} - 6787159709x^{4} + 1705354273x^{3} + 94640317x^{2} - 52200870x - 2944934

Atkin–Lehner signs

ppwpw_p
181βˆ’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’24βˆ’16βˆ’73βˆ’244βˆ’96βˆ’30βˆ’288βˆ’648βˆ’774βˆ’334160βˆ’1742βˆ’836βˆ’458319βˆ’972384βˆ’507βˆ’27761638βˆ’2150βˆ’164420868

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 0.132 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, 181]: [1, 47] (certified at ell = 2147482801)
  • T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 7 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(181, 4)[1]   # 181.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 181:4 1000