🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 881 β€Ί 881.2.a β€Ί 881.2.a.b

Newform orbit 881.2.a.b

Trace form

The 46 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

46q+4q2+7q3+64q4+5q6+31q7+12q8+57q9+19q10+12q11+12q12+9q13+5q14+23q15+96q16βˆ’3q17+12q18+33q19βˆ’16q20+13q21+19q22+16q23βˆ’7q24+O(q25)46q + 4q^{2} + 7q^{3} + 64q^{4} + 5q^{6} + 31q^{7} + 12q^{8} + 57q^{9} + 19q^{10} + 12q^{11} + 12q^{12} + 9q^{13} + 5q^{14} + 23q^{15} + 96q^{16} - 3q^{17} + 12q^{18} + 33q^{19} - 16q^{20} + 13q^{21} + 19q^{22} + 16q^{23} - 7q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 46. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x46βˆ’11x45βˆ’120x44+1686x43+5286x42βˆ’115018x41βˆ’63205x40+4637379x39βˆ’3455743x38βˆ’123837396x37+189221576x36+2327878086x35βˆ’4787631423x34βˆ’31977272980x33+77945820511x32+329102630684x31βˆ’894059846185x30βˆ’2582974094276x29+7513129274147x28+15667017946936x27βˆ’47126703963819x26βˆ’74209963900642x25+222276890368033x24+276662429815853x23βˆ’787715469635410x22βˆ’814912891864653x21+2080881888443241x20+1889843003739534x19βˆ’4034819570409444x18βˆ’3398648446036471x17+5604766587637319x16+4598841837584354x15βˆ’5385430926426912x14βˆ’4472865456791658x13+3416140415213301x12+2950151049336420x11βˆ’1358066796679579x10βˆ’1241965167746681x9+326210898146953x8+318829608744981x7βˆ’46089674878804x6βˆ’47229945694618x5+3900688213958x4+3625527372656x3βˆ’235070917653x2βˆ’109157340571x+9760252369x^{46} - 11x^{45} - 120x^{44} + 1686x^{43} + 5286x^{42} - 115018x^{41} - 63205x^{40} + 4637379x^{39} - 3455743x^{38} - 123837396x^{37} + 189221576x^{36} + 2327878086x^{35} - 4787631423x^{34} - 31977272980x^{33} + 77945820511x^{32} + 329102630684x^{31} - 894059846185x^{30} - 2582974094276x^{29} + 7513129274147x^{28} + 15667017946936x^{27} - 47126703963819x^{26} - 74209963900642x^{25} + 222276890368033x^{24} + 276662429815853x^{23} - 787715469635410x^{22} - 814912891864653x^{21} + 2080881888443241x^{20} + 1889843003739534x^{19} - 4034819570409444x^{18} - 3398648446036471x^{17} + 5604766587637319x^{16} + 4598841837584354x^{15} - 5385430926426912x^{14} - 4472865456791658x^{13} + 3416140415213301x^{12} + 2950151049336420x^{11} - 1358066796679579x^{10} - 1241965167746681x^{9} + 326210898146953x^{8} + 318829608744981x^{7} - 46089674878804x^{6} - 47229945694618x^{5} + 3900688213958x^{4} + 3625527372656x^{3} - 235070917653x^{2} - 109157340571x + 9760252369

Atkin–Lehner signs

ppwpw_p
881βˆ’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=βˆ’1w_N = -1, so the sign of the functional equation is (βˆ’1)k/2wN=+1(-1)^{k/2} w_N = +1: each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

pp2357111317192329313741434753596167717379838997
tr⁑ap\operatorname{tr} a_p47031129βˆ’33316188013βˆ’3434βˆ’10745305915212βˆ’12βˆ’31βˆ’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 0.28 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, 881]: [0, 74] (certified at ell = 2147482481)
  • T = [(2, 1), (3, 1)]; h of degree 73 = 1 x 73 (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(881, 2)[1]   # 881.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 881:2 1000