🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 647 β€Ί 647.2.a β€Ί 647.2.a.b

Newform orbit 647.2.a.b

Trace form

The 14 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

14qβˆ’2q2βˆ’4q3+4q4βˆ’5q5βˆ’9q6βˆ’9q7βˆ’6q8βˆ’6q9βˆ’7q10βˆ’5q11+7q12βˆ’25q13βˆ’3q14βˆ’5q15βˆ’12q16βˆ’18q17βˆ’11q18βˆ’2q19βˆ’7q20βˆ’4q21βˆ’10q22βˆ’12q23βˆ’18q24+O(q25)14q - 2q^{2} - 4q^{3} + 4q^{4} - 5q^{5} - 9q^{6} - 9q^{7} - 6q^{8} - 6q^{9} - 7q^{10} - 5q^{11} + 7q^{12} - 25q^{13} - 3q^{14} - 5q^{15} - 12q^{16} - 18q^{17} - 11q^{18} - 2q^{19} - 7q^{20} - 4q^{21} - 10q^{22} - 12q^{23} - 18q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 14. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x14+6x13βˆ’7x12βˆ’82x11βˆ’13x10+383x9+164x8βˆ’705x7βˆ’348x6+450x5+254x4βˆ’51x3βˆ’32x2+2x+1x^{14} + 6x^{13} - 7x^{12} - 82x^{11} - 13x^{10} + 383x^{9} + 164x^{8} - 705x^{7} - 348x^{6} + 450x^{5} + 254x^{4} - 51x^{3} - 32x^{2} + 2x + 1

Atkin–Lehner signs

ppwpw_p
647+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’2βˆ’4βˆ’5βˆ’9βˆ’5βˆ’25βˆ’18βˆ’2βˆ’120βˆ’16βˆ’3212βˆ’39βˆ’5βˆ’15βˆ’5βˆ’3βˆ’2812βˆ’55βˆ’513βˆ’24βˆ’40

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 0.225 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, 647]: [0, 55] (certified at ell = 2147482943)
  • 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(647, 2)[1]   # 647.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 647:2 1000