🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 659 β€Ί 659.2.a β€Ί 659.2.a.c

Newform orbit 659.2.a.c

Trace form

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

16qβˆ’3q2βˆ’9q3+5q4βˆ’8q5βˆ’7q6βˆ’3q7βˆ’q9βˆ’7q10βˆ’8q11βˆ’11q12βˆ’29q13βˆ’q14+8q15βˆ’13q16βˆ’2q17βˆ’26q19βˆ’10q20βˆ’15q21βˆ’11q22+q23βˆ’4q24+O(q25)16q - 3q^{2} - 9q^{3} + 5q^{4} - 8q^{5} - 7q^{6} - 3q^{7} - q^{9} - 7q^{10} - 8q^{11} - 11q^{12} - 29q^{13} - q^{14} + 8q^{15} - 13q^{16} - 2q^{17} - 26q^{19} - 10q^{20} - 15q^{21} - 11q^{22} + q^{23} - 4q^{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+3T3T = T_{2} + 3T_{3}, whose minimal polynomial is

x16+30x15+241x14βˆ’1031x13βˆ’22460x12βˆ’54560x11+491119x10+2168607x9βˆ’4283795x8βˆ’27837029x7+18529997x6+166096257x5βˆ’71678991x4βˆ’484067682x3+268763001x2+554152139xβˆ’430265413x^{16} + 30x^{15} + 241x^{14} - 1031x^{13} - 22460x^{12} - 54560x^{11} + 491119x^{10} + 2168607x^{9} - 4283795x^{8} - 27837029x^{7} + 18529997x^{6} + 166096257x^{5} - 71678991x^{4} - 484067682x^{3} + 268763001x^{2} + 554152139x - 430265413

Atkin–Lehner signs

ppwpw_p
659+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’8βˆ’3βˆ’8βˆ’29βˆ’2βˆ’261βˆ’18βˆ’28βˆ’15βˆ’240βˆ’6βˆ’39βˆ’80βˆ’192βˆ’31βˆ’1416βˆ’8βˆ’6

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 0.233 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, 659]: [0, 56] (certified at ell = 2147471831)
  • T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 8 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(659, 2)[2]   # 659.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 659:2 1000