🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 877 β€Ί 877.2.a β€Ί 877.2.a.b

Newform orbit 877.2.a.b

Trace form

The 32 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

32qβˆ’9q2βˆ’20q3+25q4βˆ’19q5+3q6βˆ’17q7βˆ’27q8+26q9+7q10βˆ’38q11βˆ’26q12βˆ’9q13βˆ’29q14βˆ’15q15+31q16βˆ’29q17βˆ’33q18βˆ’11q19βˆ’55q20+21q21+q22βˆ’36q23βˆ’6q24+O(q25)32q - 9q^{2} - 20q^{3} + 25q^{4} - 19q^{5} + 3q^{6} - 17q^{7} - 27q^{8} + 26q^{9} + 7q^{10} - 38q^{11} - 26q^{12} - 9q^{13} - 29q^{14} - 15q^{15} + 31q^{16} - 29q^{17} - 33q^{18} - 11q^{19} - 55q^{20} + 21q^{21} + q^{22} - 36q^{23} - 6q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 32. It is generated by the eigenvalue Ξ±\alpha of T=T2+3T3T = T_{2} + 3T_{3}, whose minimal polynomial is

x32+69x31+1778x30+15549x29βˆ’168426x28βˆ’4695025x27βˆ’22483660x26+321128808x25+4023231802x24βˆ’370038111x23βˆ’234499880248x22βˆ’914194456414x21+6308177209263x20+48925343044830x19βˆ’57041665175976x18βˆ’1269326895735559x17βˆ’1085930380057458x16+19209179682916951x15+36998562952447491x14βˆ’181043131826930010x13βˆ’460857463971473688x12+1126197392760259510x11+3038270159686244984x10βˆ’4989272508666355713x9βˆ’10583344338027600658x8+16021349428423346191x7+14776425462538168368x6βˆ’29234617148247930805x5+5136218994186844022x4+8866361561418472324x3βˆ’3306847657097582439x2βˆ’499830006129990832x+251207100080882609x^{32} + 69x^{31} + 1778x^{30} + 15549x^{29} - 168426x^{28} - 4695025x^{27} - 22483660x^{26} + 321128808x^{25} + 4023231802x^{24} - 370038111x^{23} - 234499880248x^{22} - 914194456414x^{21} + 6308177209263x^{20} + 48925343044830x^{19} - 57041665175976x^{18} - 1269326895735559x^{17} - 1085930380057458x^{16} + 19209179682916951x^{15} + 36998562952447491x^{14} - 181043131826930010x^{13} - 460857463971473688x^{12} + 1126197392760259510x^{11} + 3038270159686244984x^{10} - 4989272508666355713x^{9} - 10583344338027600658x^{8} + 16021349428423346191x^{7} + 14776425462538168368x^{6} - 29234617148247930805x^{5} + 5136218994186844022x^{4} + 8866361561418472324x^{3} - 3306847657097582439x^{2} - 499830006129990832x + 251207100080882609

Atkin–Lehner signs

ppwpw_p
877+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’9βˆ’20βˆ’19βˆ’17βˆ’38βˆ’9βˆ’29βˆ’11βˆ’368βˆ’19βˆ’25βˆ’18βˆ’28βˆ’73βˆ’34βˆ’89βˆ’16βˆ’24βˆ’47βˆ’6βˆ’3βˆ’122βˆ’52βˆ’4

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 0.295 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, 877]: [0, 73] (certified at ell = 2147478961)
  • T = [(2, 1), (3, 3)]; h of degree 72 = 1 x 72 (phi(ord eps) x dim S_k^new); 10 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(877, 2)[1]   # 877.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 877:2 1000