🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 641 β€Ί 641.2.a β€Ί 641.2.a.a

Newform orbit 641.2.a.a

Trace form

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

20qβˆ’3q2βˆ’9q3+9q4βˆ’5q5βˆ’11q6βˆ’24q7βˆ’3q8+7q9βˆ’17q10βˆ’14q11βˆ’10q12βˆ’14q13βˆ’8q14βˆ’14q15βˆ’9q16βˆ’11q17βˆ’3q18βˆ’38q19βˆ’11q20βˆ’7q21βˆ’8q22βˆ’11q23βˆ’23q24+O(q25)20q - 3q^{2} - 9q^{3} + 9q^{4} - 5q^{5} - 11q^{6} - 24q^{7} - 3q^{8} + 7q^{9} - 17q^{10} - 14q^{11} - 10q^{12} - 14q^{13} - 8q^{14} - 14q^{15} - 9q^{16} - 11q^{17} - 3q^{18} - 38q^{19} - 11q^{20} - 7q^{21} - 8q^{22} - 11q^{23} - 23q^{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+12x19+25x18βˆ’205x17βˆ’832x16+998x15+7346x14βˆ’1141x13βˆ’31037x12βˆ’1441x11+71812x10βˆ’3899x9βˆ’88679x8+23245x7+46845x6βˆ’21549x5βˆ’3473x4+1884x3+109x2βˆ’29x+1x^{20} + 12x^{19} + 25x^{18} - 205x^{17} - 832x^{16} + 998x^{15} + 7346x^{14} - 1141x^{13} - 31037x^{12} - 1441x^{11} + 71812x^{10} - 3899x^{9} - 88679x^{8} + 23245x^{7} + 46845x^{6} - 21549x^{5} - 3473x^{4} + 1884x^{3} + 109x^{2} - 29x + 1

Atkin–Lehner signs

ppwpw_p
641+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’3βˆ’9βˆ’5βˆ’24βˆ’14βˆ’14βˆ’11βˆ’38βˆ’11βˆ’12βˆ’65βˆ’5βˆ’17βˆ’7βˆ’32βˆ’1βˆ’55βˆ’3βˆ’29βˆ’28βˆ’9223βˆ’65

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 0.22 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, 641]: [0, 54] (certified at ell = 2147479681)
  • T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (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(641, 2)[0]   # 641.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 641:2 1000