🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 923 β€Ί 923.2.a β€Ί 923.2.a.c

Newform orbit 923.2.a.c

Trace form

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

16qβˆ’4q2βˆ’4q3+10q4βˆ’10q5βˆ’5q6βˆ’13q7βˆ’12q8+12q9βˆ’7q10βˆ’q11βˆ’2q12βˆ’16q13βˆ’3q14βˆ’11q15+6q16+q17βˆ’9q18βˆ’15q19βˆ’3q20βˆ’22q21βˆ’15q22βˆ’12q23βˆ’6q24+O(q25)16q - 4q^{2} - 4q^{3} + 10q^{4} - 10q^{5} - 5q^{6} - 13q^{7} - 12q^{8} + 12q^{9} - 7q^{10} - q^{11} - 2q^{12} - 16q^{13} - 3q^{14} - 11q^{15} + 6q^{16} + q^{17} - 9q^{18} - 15q^{19} - 3q^{20} - 22q^{21} - 15q^{22} - 12q^{23} - 6q^{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+8x15βˆ’14x14βˆ’230x13βˆ’96x12+2444x11+2301x10βˆ’12583x9βˆ’12982x8+32937x7+30667x6βˆ’40584x5βˆ’29767x4+17897x3+8468x2+852x+8x^{16} + 8x^{15} - 14x^{14} - 230x^{13} - 96x^{12} + 2444x^{11} + 2301x^{10} - 12583x^{9} - 12982x^{8} + 32937x^{7} + 30667x^{6} - 40584x^{5} - 29767x^{4} + 17897x^{3} + 8468x^{2} + 852x + 8

Atkin–Lehner signs

ppwpw_p
13+1+1
71+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’4βˆ’4βˆ’10βˆ’13βˆ’1βˆ’161βˆ’15βˆ’12βˆ’17βˆ’18βˆ’69βˆ’6βˆ’1214βˆ’343βˆ’13βˆ’49βˆ’16βˆ’45βˆ’2021βˆ’13βˆ’69

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 0.269 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, 13, 71, 923]: [0, 1, 7, 86] (certified at ell = 2147482681)
  • T = [(2, 1), (3, 1)]; h of degree 71 = 1 x 71 (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(923, 2)[2]   # 923.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 923:2 1000