🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 921 β€Ί 921.2.a β€Ί 921.2.a.g

Newform orbit 921.2.a.g

Trace form

The 13 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

13qβˆ’2q2βˆ’13q3+12q4βˆ’9q5+2q6βˆ’8q7βˆ’3q8+13q9βˆ’9q10βˆ’16q11βˆ’12q12+8q13βˆ’12q14+9q15+10q16βˆ’8q17βˆ’2q18βˆ’17q19βˆ’32q20+8q21+4q22βˆ’18q23+3q24+O(q25)13q - 2q^{2} - 13q^{3} + 12q^{4} - 9q^{5} + 2q^{6} - 8q^{7} - 3q^{8} + 13q^{9} - 9q^{10} - 16q^{11} - 12q^{12} + 8q^{13} - 12q^{14} + 9q^{15} + 10q^{16} - 8q^{17} - 2q^{18} - 17q^{19} - 32q^{20} + 8q^{21} + 4q^{22} - 18q^{23} + 3q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 13. It is generated by the eigenvalue Ξ±\alpha of T=T2+T5T = T_{2} + T_{5}, whose minimal polynomial is

x13+11x12+7x11βˆ’252x10βˆ’569x9+1721x8+5530x7βˆ’2698x6βˆ’17795x5βˆ’5136x4+18358x3+8584x2βˆ’6160xβˆ’2560x^{13} + 11x^{12} + 7x^{11} - 252x^{10} - 569x^{9} + 1721x^{8} + 5530x^{7} - 2698x^{6} - 17795x^{5} - 5136x^{4} + 18358x^{3} + 8584x^{2} - 6160x - 2560

Atkin–Lehner signs

ppwpw_p
3+1+1
307+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’13βˆ’9βˆ’8βˆ’168βˆ’8βˆ’17βˆ’18βˆ’20βˆ’4227βˆ’247βˆ’29βˆ’16βˆ’345βˆ’2βˆ’6711βˆ’25βˆ’13βˆ’447

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 0.275 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, 3, 307, 921]: [0, 1, 26, 104] (certified at ell = 2147478013)
  • T = [(2, 1), (5, 1)]; h of degree 51 = 1 x 51 (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(921, 2)[6]   # 921.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 921:2 1000