🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 463 β€Ί 463.2.a β€Ί 463.2.a.a

Newform orbit 463.2.a.a

Trace form

The 16 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

16qβˆ’9q2βˆ’6q3+15q4βˆ’16q5βˆ’6q6βˆ’10q7βˆ’24q8+8q9+3q10βˆ’7q11βˆ’9q12βˆ’17q13βˆ’4q14βˆ’7q15+21q16βˆ’46q17βˆ’16q18βˆ’7q19βˆ’28q20βˆ’13q21+q22βˆ’16q23βˆ’5q24+O(q25)16q - 9q^{2} - 6q^{3} + 15q^{4} - 16q^{5} - 6q^{6} - 10q^{7} - 24q^{8} + 8q^{9} + 3q^{10} - 7q^{11} - 9q^{12} - 17q^{13} - 4q^{14} - 7q^{15} + 21q^{16} - 46q^{17} - 16q^{18} - 7q^{19} - 28q^{20} - 13q^{21} + q^{22} - 16q^{23} - 5q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 16. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x16+15x15+67x14βˆ’17x13βˆ’809x12βˆ’1174x11+3046x10+6996x9βˆ’3987x8βˆ’15289x7βˆ’515x6+13822x5+4961x4βˆ’3788x3βˆ’2744x2βˆ’573xβˆ’37x^{16} + 15x^{15} + 67x^{14} - 17x^{13} - 809x^{12} - 1174x^{11} + 3046x^{10} + 6996x^{9} - 3987x^{8} - 15289x^{7} - 515x^{6} + 13822x^{5} + 4961x^{4} - 3788x^{3} - 2744x^{2} - 573x - 37

Atkin–Lehner signs

ppwpw_p
463+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’6βˆ’16βˆ’10βˆ’7βˆ’17βˆ’46βˆ’7βˆ’16βˆ’1041βˆ’5610βˆ’28βˆ’36βˆ’74011βˆ’2015βˆ’41βˆ’73βˆ’45

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 0.189 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, 463]: [0, 39] (certified at ell = 2147483647)
  • T = [(2, 1), (3, 1)]; h of degree 38 = 1 x 38 (phi(ord eps) x dim S_k^new); 4 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(463, 2)[0]   # 463.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 463:2 1000