🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 227 β€Ί 227.4.a β€Ί 227.4.a.a

Newform orbit 227.4.a.a

Trace form

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

23qβˆ’7q2βˆ’17q3+65q4βˆ’41q5βˆ’51q6βˆ’86q7βˆ’60q8+124q9βˆ’82q10βˆ’99q11βˆ’216q12βˆ’315q13βˆ’56q14βˆ’184q15βˆ’51q16βˆ’176q17βˆ’306q18βˆ’117q19βˆ’480q20βˆ’324q21βˆ’717q22βˆ’276q23βˆ’510q24+O(q25)23q - 7q^{2} - 17q^{3} + 65q^{4} - 41q^{5} - 51q^{6} - 86q^{7} - 60q^{8} + 124q^{9} - 82q^{10} - 99q^{11} - 216q^{12} - 315q^{13} - 56q^{14} - 184q^{15} - 51q^{16} - 176q^{17} - 306q^{18} - 117q^{19} - 480q^{20} - 324q^{21} - 717q^{22} - 276q^{23} - 510q^{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+T3T = T_{2} + T_{3}, whose minimal polynomial is

x23+24x22βˆ’158x21βˆ’6995x20βˆ’918x19+850610x18+1527423x17βˆ’57416830x16βˆ’118293321x15+2369651095x14+4069892303x13βˆ’61316445425x12βˆ’69403558414x11+982562164231x10+492373863223x9βˆ’9348643930946x8+792054327783x7+48224567356511x6βˆ’28232072864650x5βˆ’107337591598653x4+106138238869372x3+32127413146550x2βˆ’43191802541404x+1916228672804x^{23} + 24x^{22} - 158x^{21} - 6995x^{20} - 918x^{19} + 850610x^{18} + 1527423x^{17} - 57416830x^{16} - 118293321x^{15} + 2369651095x^{14} + 4069892303x^{13} - 61316445425x^{12} - 69403558414x^{11} + 982562164231x^{10} + 492373863223x^{9} - 9348643930946x^{8} + 792054327783x^{7} + 48224567356511x^{6} - 28232072864650x^{5} - 107337591598653x^{4} + 106138238869372x^{3} + 32127413146550x^{2} - 43191802541404x + 1916228672804

Atkin–Lehner signs

ppwpw_p
227βˆ’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=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’7βˆ’17βˆ’41βˆ’86βˆ’99βˆ’315βˆ’176βˆ’117βˆ’276βˆ’635βˆ’642βˆ’2463βˆ’464βˆ’1257βˆ’548βˆ’20271354βˆ’693βˆ’22691288βˆ’2400βˆ’14641151470βˆ’2782

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 0.155 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 engine's checks
  • dim M^+ at levels [1, 227]: [1, 58] (certified at ell = 2147479573)
  • T = [(2, 1), (3, 1)]; h of degree 56 = 1 x 56 (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(227, 4)[0]   # 227.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 227:4 1000