🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 901 β€Ί 901.2.a β€Ί 901.2.a.g

Newform orbit 901.2.a.g

Trace form

The 12 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

12qβˆ’q2βˆ’3q3+9q4βˆ’5q5βˆ’5q6+3q7βˆ’21q8+3q9+2q10βˆ’20q11+2q12βˆ’16q13βˆ’12q14βˆ’17q15βˆ’5q16βˆ’12q17βˆ’9q18βˆ’17q19βˆ’9q20βˆ’3q21+2q22βˆ’11q23βˆ’14q24+O(q25)12q - q^{2} - 3q^{3} + 9q^{4} - 5q^{5} - 5q^{6} + 3q^{7} - 21q^{8} + 3q^{9} + 2q^{10} - 20q^{11} + 2q^{12} - 16q^{13} - 12q^{14} - 17q^{15} - 5q^{16} - 12q^{17} - 9q^{18} - 17q^{19} - 9q^{20} - 3q^{21} + 2q^{22} - 11q^{23} - 14q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 12. It is generated by the eigenvalue Ξ±\alpha of T=T2+3T3+5T5+7T7T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}, whose minimal polynomial is

x12+14x11βˆ’1318x10βˆ’25610x9+446755x8+14022557x7+36412095x6βˆ’2058132506x5βˆ’27078938060x4βˆ’142700839210x3βˆ’330194258349x2βˆ’229655245896x+101954657884x^{12} + 14x^{11} - 1318x^{10} - 25610x^{9} + 446755x^{8} + 14022557x^{7} + 36412095x^{6} - 2058132506x^{5} - 27078938060x^{4} - 142700839210x^{3} - 330194258349x^{2} - 229655245896x + 101954657884

Atkin–Lehner signs

ppwpw_p
17+1+1
53+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’1βˆ’3βˆ’53βˆ’20βˆ’16βˆ’12βˆ’17βˆ’11βˆ’816βˆ’8βˆ’19βˆ’19βˆ’12βˆ’52βˆ’60βˆ’2120βˆ’27βˆ’18βˆ’232

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 0.418 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, 17, 53, 901]: [0, 2, 5, 82] (certified at ell = 2147482481)
  • T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 69 = 1 x 69 (phi(ord eps) x dim S_k^new); 15 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(901, 2)[6]   # 901.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 901:2 1000