🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 919 β€Ί 919.2.a β€Ί 919.2.a.c

Newform orbit 919.2.a.c

Trace form

The 47 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

47q+5q2+9q3+51q4+22q5+9q6βˆ’2q7+12q8+80q9βˆ’4q10+11q11+20q12+11q13+31q14+4q15+51q16+53q17+6q18+7q19+41q20+20q21βˆ’12q22+14q23+12q24+O(q25)47q + 5q^{2} + 9q^{3} + 51q^{4} + 22q^{5} + 9q^{6} - 2q^{7} + 12q^{8} + 80q^{9} - 4q^{10} + 11q^{11} + 20q^{12} + 11q^{13} + 31q^{14} + 4q^{15} + 51q^{16} + 53q^{17} + 6q^{18} + 7q^{19} + 41q^{20} + 20q^{21} - 12q^{22} + 14q^{23} + 12q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 47. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x47βˆ’14x46βˆ’94x45+2131x44+901x43βˆ’142636x42+264778x41+5547552x40βˆ’17949039x39βˆ’139228746x38+613676418x37+2357005260x36βˆ’13539700330x35βˆ’27038178582x34+209688242399x33+198964425399x32βˆ’2376053216862x31βˆ’694439753070x30+20171939610946x29βˆ’2825318829408x28βˆ’130146287779632x27+55808852315424x26+643425902609305x25βˆ’396850769674481x24βˆ’2446845543022355x23+1781287739558486x22+7156435274622321x21βˆ’5557637902796709x20βˆ’16029439289600923x19+12366526974203900x18+27247057315711126x17βˆ’19599237348236324x16βˆ’34615771752654148x15+21689487780174367x14+32100340707856563x13βˆ’16094632059867808x12βˆ’20960547634786396x11+7439194816633124x10+9111443162087001x9βˆ’1845492449419684x8βˆ’2396763376053722x7+154453946397152x6+312771039030282x5+10008189929331x4βˆ’10668133371154x3+392821967083x2+21411518790xβˆ’340532271x^{47} - 14x^{46} - 94x^{45} + 2131x^{44} + 901x^{43} - 142636x^{42} + 264778x^{41} + 5547552x^{40} - 17949039x^{39} - 139228746x^{38} + 613676418x^{37} + 2357005260x^{36} - 13539700330x^{35} - 27038178582x^{34} + 209688242399x^{33} + 198964425399x^{32} - 2376053216862x^{31} - 694439753070x^{30} + 20171939610946x^{29} - 2825318829408x^{28} - 130146287779632x^{27} + 55808852315424x^{26} + 643425902609305x^{25} - 396850769674481x^{24} - 2446845543022355x^{23} + 1781287739558486x^{22} + 7156435274622321x^{21} - 5557637902796709x^{20} - 16029439289600923x^{19} + 12366526974203900x^{18} + 27247057315711126x^{17} - 19599237348236324x^{16} - 34615771752654148x^{15} + 21689487780174367x^{14} + 32100340707856563x^{13} - 16094632059867808x^{12} - 20960547634786396x^{11} + 7439194816633124x^{10} + 9111443162087001x^{9} - 1845492449419684x^{8} - 2396763376053722x^{7} + 154453946397152x^{6} + 312771039030282x^{5} + 10008189929331x^{4} - 10668133371154x^{3} + 392821967083x^{2} + 21411518790x - 340532271

Atkin–Lehner signs

ppwpw_p
919βˆ’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_p5922βˆ’21111537145955102βˆ’1627528623βˆ’20414βˆ’382119410

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 0.384 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, 919]: [0, 77] (certified at ell = 2147470057)
  • T = [(2, 1), (3, 1)]; h of degree 76 = 1 x 76 (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(919, 2)[2]   # 919.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 919:2 1000