🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 761 β€Ί 761.2.a β€Ί 761.2.a.b

Newform orbit 761.2.a.b

Trace form

The 20 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

20qβˆ’2q2βˆ’5q3+8q4βˆ’8q5βˆ’5q6βˆ’20q7βˆ’15q8+9q9βˆ’10q10βˆ’12q11βˆ’7q12βˆ’14q13βˆ’7q14βˆ’36q15βˆ’16q16βˆ’4q17βˆ’12q18βˆ’31q19βˆ’5q20βˆ’10q21βˆ’13q22βˆ’11q23βˆ’8q24+O(q25)20q - 2q^{2} - 5q^{3} + 8q^{4} - 8q^{5} - 5q^{6} - 20q^{7} - 15q^{8} + 9q^{9} - 10q^{10} - 12q^{11} - 7q^{12} - 14q^{13} - 7q^{14} - 36q^{15} - 16q^{16} - 4q^{17} - 12q^{18} - 31q^{19} - 5q^{20} - 10q^{21} - 13q^{22} - 11q^{23} - 8q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 20. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x20+7x19βˆ’29x18βˆ’264x17+275x16+3850x15βˆ’1352x14βˆ’29147x13+8295x12+124580x11βˆ’56102x10βˆ’294495x9+202116x8+348759x7βˆ’340541x6βˆ’142767x5+233484x4βˆ’31191x3βˆ’32372x2+5136x+1753x^{20} + 7x^{19} - 29x^{18} - 264x^{17} + 275x^{16} + 3850x^{15} - 1352x^{14} - 29147x^{13} + 8295x^{12} + 124580x^{11} - 56102x^{10} - 294495x^{9} + 202116x^{8} + 348759x^{7} - 340541x^{6} - 142767x^{5} + 233484x^{4} - 31191x^{3} - 32372x^{2} + 5136x + 1753

Atkin–Lehner signs

ppwpw_p
761+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’2βˆ’5βˆ’8βˆ’20βˆ’12βˆ’14βˆ’4βˆ’31βˆ’11βˆ’15βˆ’56βˆ’33βˆ’15βˆ’18βˆ’932βˆ’353βˆ’56βˆ’3727βˆ’90βˆ’717βˆ’8

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 0.253 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, 761]: [0, 64] (certified at ell = 2147466641)
  • T = [(2, 1), (3, 1)]; h of degree 63 = 1 x 63 (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(761, 2)[1]   # 761.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 761:2 1000