🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 799 β€Ί 799.2.a β€Ί 799.2.a.e

Newform orbit 799.2.a.e

Trace form

The 12 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

12qβˆ’4q2βˆ’q3+8q4βˆ’11q5βˆ’q6+q7βˆ’12q8βˆ’3q9βˆ’2q10βˆ’6q11βˆ’4q12βˆ’13q13βˆ’15q14βˆ’15q15+4q16βˆ’12q17βˆ’9q18βˆ’9q19βˆ’23q20βˆ’6q21+11q22βˆ’2q23βˆ’6q24+O(q25)12q - 4q^{2} - q^{3} + 8q^{4} - 11q^{5} - q^{6} + q^{7} - 12q^{8} - 3q^{9} - 2q^{10} - 6q^{11} - 4q^{12} - 13q^{13} - 15q^{14} - 15q^{15} + 4q^{16} - 12q^{17} - 9q^{18} - 9q^{19} - 23q^{20} - 6q^{21} + 11q^{22} - 2q^{23} - 6q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 12. It is generated by the eigenvalue Ξ±\alpha of T=T2+3T3+5T5+7T7+9T11T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}, whose minimal polynomial is

x12+109x11+806x10βˆ’247804x9βˆ’6179857x8+151199376x7+5618616209x6βˆ’11349241052x5βˆ’1507416977731x4βˆ’8353576311653x3+44415857678457x2+285273914102020xβˆ’19979807770049x^{12} + 109x^{11} + 806x^{10} - 247804x^{9} - 6179857x^{8} + 151199376x^{7} + 5618616209x^{6} - 11349241052x^{5} - 1507416977731x^{4} - 8353576311653x^{3} + 44415857678457x^{2} + 285273914102020x - 19979807770049

Atkin–Lehner signs

ppwpw_p
17+1+1
47+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’1βˆ’111βˆ’6βˆ’13βˆ’12βˆ’9βˆ’2βˆ’14βˆ’188βˆ’43βˆ’7βˆ’125βˆ’25βˆ’1βˆ’2βˆ’56βˆ’10βˆ’45βˆ’65βˆ’21

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 0.41 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, 17, 47, 799]: [0, 2, 5, 74] (certified at ell = 2147483137)
  • T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 61 = 1 x 61 (phi(ord eps) x dim S_k^new); 15 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(799, 2)[4]   # 799.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 799:2 1000