🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 229 β€Ί 229.4.a β€Ί 229.4.a.a

Newform orbit 229.4.a.a

Trace form

The 26 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

26qβˆ’15q2βˆ’17q3+89q4βˆ’30q5βˆ’53q6βˆ’55q7βˆ’177q8+113q9βˆ’87q10βˆ’335q11βˆ’160q12βˆ’91q13βˆ’183q14βˆ’220q15+177q16βˆ’155q17βˆ’249q18βˆ’439q19βˆ’368q20βˆ’352q21βˆ’163q22βˆ’345q23βˆ’541q24+O(q25)26q - 15q^{2} - 17q^{3} + 89q^{4} - 30q^{5} - 53q^{6} - 55q^{7} - 177q^{8} + 113q^{9} - 87q^{10} - 335q^{11} - 160q^{12} - 91q^{13} - 183q^{14} - 220q^{15} + 177q^{16} - 155q^{17} - 249q^{18} - 439q^{19} - 368q^{20} - 352q^{21} - 163q^{22} - 345q^{23} - 541q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 26. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x26+32x25+9x24βˆ’9031x23βˆ’56608x22+1055910x21+9552425x20βˆ’67744462x19βˆ’762511015x18+2702035176x17+35594895865x16βˆ’73791244712x15βˆ’1042378665363x14+1510192793863x13+19523098258117x12βˆ’23995557195654x11βˆ’230672022482177x10+276272453127444x9+1643346858747103x8βˆ’2016228341466699x7βˆ’6493387171681120x6+8062436773093723x5+12195194822001421x4βˆ’14877244708978490x3βˆ’7194929158026650x2+9585169902402500xβˆ’1700449901125000x^{26} + 32x^{25} + 9x^{24} - 9031x^{23} - 56608x^{22} + 1055910x^{21} + 9552425x^{20} - 67744462x^{19} - 762511015x^{18} + 2702035176x^{17} + 35594895865x^{16} - 73791244712x^{15} - 1042378665363x^{14} + 1510192793863x^{13} + 19523098258117x^{12} - 23995557195654x^{11} - 230672022482177x^{10} + 276272453127444x^{9} + 1643346858747103x^{8} - 2016228341466699x^{7} - 6493387171681120x^{6} + 8062436773093723x^{5} + 12195194822001421x^{4} - 14877244708978490x^{3} - 7194929158026650x^{2} + 9585169902402500x - 1700449901125000

Atkin–Lehner signs

ppwpw_p
229βˆ’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’15βˆ’17βˆ’30βˆ’55βˆ’335βˆ’91βˆ’155βˆ’439βˆ’345βˆ’979βˆ’591βˆ’330βˆ’1974βˆ’839βˆ’1533βˆ’1398βˆ’3156306βˆ’220βˆ’2026986βˆ’1135βˆ’2963βˆ’772βˆ’577

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 0.157 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, 229]: [1, 59] (certified at ell = 2147478649)
  • T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(229, 4)[0]   # 229.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 229:4 1000