🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 223 β€Ί 223.4.a β€Ί 223.4.a.a

Newform orbit 223.4.a.a

Trace form

The 24 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

24qβˆ’17q2βˆ’8q3+79q4βˆ’50q5βˆ’14q6βˆ’61q7βˆ’204q8+66q9βˆ’61q10βˆ’104q11βˆ’163q12βˆ’92q13βˆ’149q14βˆ’224q15+183q16βˆ’537q17βˆ’442q18βˆ’209q19βˆ’405q20βˆ’416q21βˆ’34q22βˆ’416q23+47q24+O(q25)24q - 17q^{2} - 8q^{3} + 79q^{4} - 50q^{5} - 14q^{6} - 61q^{7} - 204q^{8} + 66q^{9} - 61q^{10} - 104q^{11} - 163q^{12} - 92q^{13} - 149q^{14} - 224q^{15} + 183q^{16} - 537q^{17} - 442q^{18} - 209q^{19} - 405q^{20} - 416q^{21} - 34q^{22} - 416q^{23} + 47q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 24. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x24+25x23βˆ’166x22βˆ’7851x21βˆ’1695x20+1056578x19+2114042x18βˆ’80522809x17βˆ’184356777x16+3864419084x15+7016026540x14βˆ’121917165119x13βˆ’113929446191x12+2522800981400x11βˆ’188337143952x10βˆ’32539471931265x9+34337034410670x8+229592398119321x7βˆ’471809749664904x6βˆ’569568643687732x5+2293408602086880x4βˆ’1219531034801600x3βˆ’1755117314824448x2+1632270456201216xβˆ’6729628368896x^{24} + 25x^{23} - 166x^{22} - 7851x^{21} - 1695x^{20} + 1056578x^{19} + 2114042x^{18} - 80522809x^{17} - 184356777x^{16} + 3864419084x^{15} + 7016026540x^{14} - 121917165119x^{13} - 113929446191x^{12} + 2522800981400x^{11} - 188337143952x^{10} - 32539471931265x^{9} + 34337034410670x^{8} + 229592398119321x^{7} - 471809749664904x^{6} - 569568643687732x^{5} + 2293408602086880x^{4} - 1219531034801600x^{3} - 1755117314824448x^{2} + 1632270456201216x - 6729628368896

Atkin–Lehner signs

ppwpw_p
223βˆ’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’17βˆ’8βˆ’50βˆ’61βˆ’104βˆ’92βˆ’537βˆ’209βˆ’416βˆ’749βˆ’249βˆ’1143βˆ’1293βˆ’1027βˆ’1795βˆ’2157βˆ’1196650364βˆ’1048βˆ’25453120βˆ’1537βˆ’1835βˆ’420

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 0.151 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, 223]: [1, 57] (certified at ell = 2147482591)
  • T = [(2, 1), (3, 1)]; h of degree 55 = 1 x 55 (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(223, 4)[0]   # 223.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 223:4 1000