🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 863 β€Ί 863.2.a β€Ί 863.2.a.b

Newform orbit 863.2.a.b

Trace form

The 22 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

22qβˆ’6q2βˆ’2q3+16q4βˆ’2q5βˆ’13q6βˆ’20q7βˆ’21q8+12q9βˆ’12q10βˆ’4q11βˆ’18q12βˆ’44q13+2q14βˆ’22q15βˆ’4q16βˆ’28q17βˆ’4q18βˆ’29q19βˆ’20q21βˆ’10q22βˆ’4q23βˆ’q24+O(q25)22q - 6q^{2} - 2q^{3} + 16q^{4} - 2q^{5} - 13q^{6} - 20q^{7} - 21q^{8} + 12q^{9} - 12q^{10} - 4q^{11} - 18q^{12} - 44q^{13} + 2q^{14} - 22q^{15} - 4q^{16} - 28q^{17} - 4q^{18} - 29q^{19} - 20q^{21} - 10q^{22} - 4q^{23} - q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 22. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x22+8x21βˆ’24x20βˆ’294x19+95x18+4233x17+1956x16βˆ’30485x15βˆ’23619x14+115486x13+103037x12βˆ’221288x11βˆ’197488x10+195264x9+149773x8βˆ’84157x7βˆ’42439x6+18059x5+2876x4βˆ’1538x3+86x2+14xβˆ’1x^{22} + 8x^{21} - 24x^{20} - 294x^{19} + 95x^{18} + 4233x^{17} + 1956x^{16} - 30485x^{15} - 23619x^{14} + 115486x^{13} + 103037x^{12} - 221288x^{11} - 197488x^{10} + 195264x^{9} + 149773x^{8} - 84157x^{7} - 42439x^{6} + 18059x^{5} + 2876x^{4} - 1538x^{3} + 86x^{2} + 14x - 1

Atkin–Lehner signs

ppwpw_p
863+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’6βˆ’2βˆ’2βˆ’20βˆ’4βˆ’44βˆ’28βˆ’29βˆ’4βˆ’6βˆ’8βˆ’21βˆ’53βˆ’28βˆ’16βˆ’513βˆ’10βˆ’2021βˆ’126βˆ’2βˆ’243βˆ’64

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 0.27 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, 863]: [0, 73] (certified at ell = 2147466121)
  • T = [(2, 1), (3, 1)]; h of degree 72 = 1 x 72 (phi(ord eps) x dim S_k^new); 7 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(863, 2)[1]   # 863.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 863:2 1000