🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 173 β€Ί 173.4.a β€Ί 173.4.a.a

Newform orbit 173.4.a.a

Trace form

The 18 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

18qβˆ’7q2βˆ’28q3+49q4βˆ’25q5βˆ’12q6βˆ’93q7βˆ’63q8+114q9βˆ’126q10βˆ’94q11βˆ’236q12βˆ’132q13+9q14βˆ’184q15βˆ’47q16βˆ’88q17βˆ’426q18βˆ’348q19βˆ’472q20βˆ’108q21βˆ’577q22βˆ’392q23βˆ’227q24+O(q25)18q - 7q^{2} - 28q^{3} + 49q^{4} - 25q^{5} - 12q^{6} - 93q^{7} - 63q^{8} + 114q^{9} - 126q^{10} - 94q^{11} - 236q^{12} - 132q^{13} + 9q^{14} - 184q^{15} - 47q^{16} - 88q^{17} - 426q^{18} - 348q^{19} - 472q^{20} - 108q^{21} - 577q^{22} - 392q^{23} - 227q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 18. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x18+35x17+228x16βˆ’4365x15βˆ’54005x14+118631x13+3342910x12+2248916x11βˆ’91443107x10βˆ’142450047x9+1229512402x8+2123210348x7βˆ’7696323522x6βˆ’11074233170x5+16551707926x4+9567556858x3βˆ’4037633625x2βˆ’433550994x+104998656x^{18} + 35x^{17} + 228x^{16} - 4365x^{15} - 54005x^{14} + 118631x^{13} + 3342910x^{12} + 2248916x^{11} - 91443107x^{10} - 142450047x^{9} + 1229512402x^{8} + 2123210348x^{7} - 7696323522x^{6} - 11074233170x^{5} + 16551707926x^{4} + 9567556858x^{3} - 4037633625x^{2} - 433550994x + 104998656

Atkin–Lehner signs

ppwpw_p
173βˆ’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’28βˆ’25βˆ’93βˆ’94βˆ’132βˆ’88βˆ’348βˆ’392βˆ’414βˆ’734βˆ’864βˆ’187βˆ’25036322231264βˆ’21βˆ’4255βˆ’663βˆ’1957βˆ’2007βˆ’12091835βˆ’810

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 0.125 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, 173]: [1, 45] (certified at ell = 2147482093)
  • T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 6 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(173, 4)[0]   # 173.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 173:4 1000