🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 109 β€Ί 109.4.a β€Ί 109.4.a.a

Newform orbit 109.4.a.a

Trace form

The 12 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

12qβˆ’6q2βˆ’19q3+36q4βˆ’40q5βˆ’53q6βˆ’32q7βˆ’69q8+55q9βˆ’64q10βˆ’206q11βˆ’192q12βˆ’89q13βˆ’289q14βˆ’138q15+56q16βˆ’140q17βˆ’52q18βˆ’332q19βˆ’555q20βˆ’176q21+27q22βˆ’351q23βˆ’635q24+O(q25)12q - 6q^{2} - 19q^{3} + 36q^{4} - 40q^{5} - 53q^{6} - 32q^{7} - 69q^{8} + 55q^{9} - 64q^{10} - 206q^{11} - 192q^{12} - 89q^{13} - 289q^{14} - 138q^{15} + 56q^{16} - 140q^{17} - 52q^{18} - 332q^{19} - 555q^{20} - 176q^{21} + 27q^{22} - 351q^{23} - 635q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 12. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x12+25x11+110x10βˆ’1326x9βˆ’9523x8+14540x7+174000x6+5222x5βˆ’933500x4βˆ’75993x3+1035512x2βˆ’21604xβˆ’125688x^{12} + 25x^{11} + 110x^{10} - 1326x^{9} - 9523x^{8} + 14540x^{7} + 174000x^{6} + 5222x^{5} - 933500x^{4} - 75993x^{3} + 1035512x^{2} - 21604x - 125688

Atkin–Lehner signs

ppwpw_p
109βˆ’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’6βˆ’19βˆ’40βˆ’32βˆ’206βˆ’89βˆ’140βˆ’332βˆ’351βˆ’280βˆ’276205βˆ’942115βˆ’543βˆ’349βˆ’3014βˆ’264292βˆ’3172βˆ’42957βˆ’2061βˆ’1573βˆ’1077

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 0.105 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, 109]: [1, 29] (certified at ell = 2147482801)
  • T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 4 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(109, 4)[0]   # 109.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 109:4 1000