🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 661 β€Ί 661.2.a β€Ί 661.2.a.b

Newform orbit 661.2.a.b

Trace form

The 23 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

23qβˆ’9q2βˆ’7q3+15q4βˆ’11q5βˆ’10q6βˆ’6q7βˆ’24q8+10q9βˆ’50q11βˆ’6q12βˆ’9q13βˆ’12q14βˆ’20q15+7q16βˆ’11q17βˆ’14q18βˆ’27q19βˆ’20q20βˆ’19q21+16q22βˆ’23q23βˆ’18q24+O(q25)23q - 9q^{2} - 7q^{3} + 15q^{4} - 11q^{5} - 10q^{6} - 6q^{7} - 24q^{8} + 10q^{9} - 50q^{11} - 6q^{12} - 9q^{13} - 12q^{14} - 20q^{15} + 7q^{16} - 11q^{17} - 14q^{18} - 27q^{19} - 20q^{20} - 19q^{21} + 16q^{22} - 23q^{23} - 18q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 23. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x23+16x22+68x21βˆ’176x20βˆ’1922x19βˆ’1702x18+17890x17+36839x16βˆ’73752x15βˆ’232844x14+120871x13+702344x12+20875x11βˆ’1096428x10βˆ’282538x9+863795x8+276425x7βˆ’300407x6βˆ’81301x5+31666x4+8628x3βˆ’538x2βˆ’198xβˆ’9x^{23} + 16x^{22} + 68x^{21} - 176x^{20} - 1922x^{19} - 1702x^{18} + 17890x^{17} + 36839x^{16} - 73752x^{15} - 232844x^{14} + 120871x^{13} + 702344x^{12} + 20875x^{11} - 1096428x^{10} - 282538x^{9} + 863795x^{8} + 276425x^{7} - 300407x^{6} - 81301x^{5} + 31666x^{4} + 8628x^{3} - 538x^{2} - 198x - 9

Atkin–Lehner signs

ppwpw_p
661+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’9βˆ’7βˆ’11βˆ’6βˆ’50βˆ’9βˆ’11βˆ’27βˆ’23βˆ’41βˆ’14βˆ’5βˆ’19βˆ’26βˆ’33βˆ’6βˆ’85βˆ’8βˆ’18βˆ’53160βˆ’54βˆ’2931

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 0.223 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, 661]: [0, 55] (certified at ell = 2147482921)
  • T = [(2, 1), (3, 1)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 6 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(661, 2)[1]   # 661.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 661:2 1000