🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 887 β€Ί 887.2.a β€Ί 887.2.a.b

Newform orbit 887.2.a.b

Trace form

The 21 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

21qβˆ’4q2βˆ’11q3+12q4βˆ’3q5βˆ’7q6βˆ’15q7βˆ’9q8+4q9βˆ’8q10βˆ’4q11βˆ’16q12βˆ’20q13βˆ’3q14βˆ’5q15βˆ’2q16βˆ’30q17+3q18βˆ’6q19βˆ’4q20βˆ’8q21βˆ’28q22βˆ’13q23βˆ’6q24+O(q25)21q - 4q^{2} - 11q^{3} + 12q^{4} - 3q^{5} - 7q^{6} - 15q^{7} - 9q^{8} + 4q^{9} - 8q^{10} - 4q^{11} - 16q^{12} - 20q^{13} - 3q^{14} - 5q^{15} - 2q^{16} - 30q^{17} + 3q^{18} - 6q^{19} - 4q^{20} - 8q^{21} - 28q^{22} - 13q^{23} - 6q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 21. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x21+15x20+59x19βˆ’160x18βˆ’1607x17βˆ’1636x16+12445x15+28357x14βˆ’38162x13βˆ’147746x12+37240x11+386966x10+40635x9βˆ’574889x8βˆ’101287x7+490174x6+42279x5βˆ’213973x4+16869x3+30778x2βˆ’6578x+251x^{21} + 15x^{20} + 59x^{19} - 160x^{18} - 1607x^{17} - 1636x^{16} + 12445x^{15} + 28357x^{14} - 38162x^{13} - 147746x^{12} + 37240x^{11} + 386966x^{10} + 40635x^{9} - 574889x^{8} - 101287x^{7} + 490174x^{6} + 42279x^{5} - 213973x^{4} + 16869x^{3} + 30778x^{2} - 6578x + 251

Atkin–Lehner signs

ppwpw_p
887+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=+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’4βˆ’11βˆ’3βˆ’15βˆ’4βˆ’20βˆ’30βˆ’6βˆ’1302βˆ’6314βˆ’48βˆ’16βˆ’3621βˆ’25βˆ’3610βˆ’76βˆ’418βˆ’20βˆ’106

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 0.287 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 label, dimension and tr⁑ap\operatorname{tr} a_p for every p<1000p < 1000 equal LMFDB's.

The engine's checks
  • dim M^+ at levels [1, 887]: [0, 75] (certified at ell = 2147476169)
  • T = [(2, 1), (3, 1)]; h of degree 74 = 1 x 74 (phi(ord eps) x dim S_k^new); 8 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(887, 2)[1]   # 887.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 887:2 1000