🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 997 β€Ί 997.2.a β€Ί 997.2.a.g

Newform orbit 997.2.a.g

Trace form

The 23 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

23qβˆ’8q2βˆ’13q3+24q4+2q5βˆ’15q6βˆ’5q7βˆ’21q8+26q9βˆ’11q10βˆ’27q11βˆ’21q12βˆ’16q13βˆ’24q14βˆ’34q15+38q16βˆ’28q17βˆ’6q18βˆ’23q19βˆ’38q20+7q21βˆ’12q22βˆ’49q23+22q24+O(q25)23q - 8q^{2} - 13q^{3} + 24q^{4} + 2q^{5} - 15q^{6} - 5q^{7} - 21q^{8} + 26q^{9} - 11q^{10} - 27q^{11} - 21q^{12} - 16q^{13} - 24q^{14} - 34q^{15} + 38q^{16} - 28q^{17} - 6q^{18} - 23q^{19} - 38q^{20} + 7q^{21} - 12q^{22} - 49q^{23} + 22q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 23. It is generated by the eigenvalue Ξ±\alpha of T=T2+3T3+5T7+7T11+9T13T = T_{2} + 3T_{3} + 5T_{7} + 7T_{11} + 9T_{13}, whose minimal polynomial is

x23+405x22+63766x21+4281187x20βˆ’3531886x19βˆ’18611406350x18βˆ’976241406101x17+3891316282896x16+2062509547130750x15+59902489982389788x14βˆ’866625475594199083x13βˆ’80516588578807660901x12βˆ’1138406358619024116200x11+26020222770053603148628x10+1029023729849043317073241x9+8233182917660758622454143x8βˆ’146368928416988704887059229x7βˆ’3613700039610305402553900072x6βˆ’26366041839015964111793545711x5βˆ’10298667765951873050625227091x4+718012941283691268386348986770x3+1946803897938677028272522185828x2βˆ’5099900397439548830401677181088xβˆ’12082660270222699673292140335120x^{23} + 405x^{22} + 63766x^{21} + 4281187x^{20} - 3531886x^{19} - 18611406350x^{18} - 976241406101x^{17} + 3891316282896x^{16} + 2062509547130750x^{15} + 59902489982389788x^{14} - 866625475594199083x^{13} - 80516588578807660901x^{12} - 1138406358619024116200x^{11} + 26020222770053603148628x^{10} + 1029023729849043317073241x^{9} + 8233182917660758622454143x^{8} - 146368928416988704887059229x^{7} - 3613700039610305402553900072x^{6} - 26366041839015964111793545711x^{5} - 10298667765951873050625227091x^{4} + 718012941283691268386348986770x^{3} + 1946803897938677028272522185828x^{2} - 5099900397439548830401677181088x - 12082660270222699673292140335120

Atkin–Lehner signs

ppwpw_p
997+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’8βˆ’132βˆ’5βˆ’27βˆ’16βˆ’28βˆ’23βˆ’49βˆ’4βˆ’18βˆ’499βˆ’55βˆ’55βˆ’17βˆ’4βˆ’15βˆ’45βˆ’25βˆ’35βˆ’60βˆ’3513

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 0.719 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, 997]: [0, 83] (certified at ell = 2147475601)
  • T = [(2, 1), (3, 3), (7, 5), (11, 7), (13, 9)]; h of degree 82 = 1 x 82 (phi(ord eps) x dim S_k^new); 20 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(997, 2)[6]   # 997.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 997:2 1000