🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 631 β€Ί 631.2.a β€Ί 631.2.a.a

Newform orbit 631.2.a.a

Trace form

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

20qβˆ’8q2βˆ’4q3+14q4βˆ’17q5βˆ’11q6βˆ’6q7βˆ’18q8βˆ’q10βˆ’15q11βˆ’13q13βˆ’12q14βˆ’12q15+6q16βˆ’35q17βˆ’3q18βˆ’17q19βˆ’27q20βˆ’23q21+7q22βˆ’8q23βˆ’24q24+O(q25)20q - 8q^{2} - 4q^{3} + 14q^{4} - 17q^{5} - 11q^{6} - 6q^{7} - 18q^{8} - q^{10} - 15q^{11} - 13q^{13} - 12q^{14} - 12q^{15} + 6q^{16} - 35q^{17} - 3q^{18} - 17q^{19} - 27q^{20} - 23q^{21} + 7q^{22} - 8q^{23} - 24q^{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+26x18βˆ’203x17βˆ’896x16+793x15+8320x14+2525x13βˆ’36791x12βˆ’26373x11+87901x10+73605x9βˆ’113797x8βˆ’90414x7+70987x6+48258x5βˆ’13363x4βˆ’9607x3βˆ’1385x2βˆ’67xβˆ’1x^{20} + 12x^{19} + 26x^{18} - 203x^{17} - 896x^{16} + 793x^{15} + 8320x^{14} + 2525x^{13} - 36791x^{12} - 26373x^{11} + 87901x^{10} + 73605x^{9} - 113797x^{8} - 90414x^{7} + 70987x^{6} + 48258x^{5} - 13363x^{4} - 9607x^{3} - 1385x^{2} - 67x - 1

Atkin–Lehner signs

ppwpw_p
631+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’8βˆ’4βˆ’17βˆ’6βˆ’15βˆ’13βˆ’35βˆ’17βˆ’8βˆ’27βˆ’222βˆ’8510βˆ’12βˆ’24βˆ’51βˆ’714βˆ’25βˆ’26βˆ’9βˆ’7βˆ’115βˆ’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 0.217 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, 631]: [0, 53] (certified at ell = 2147478481)
  • T = [(2, 1), (3, 1)]; h of degree 52 = 1 x 52 (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(631, 2)[0]   # 631.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 631:2 1000