🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 131 β€Ί 131.4.a β€Ί 131.4.a.a

Newform orbit 131.4.a.a

Trace form

The 11 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

11qβˆ’3q2βˆ’11q3+11q4βˆ’27q5βˆ’22q6βˆ’32q7βˆ’15q8+14q9βˆ’98q10βˆ’103q11βˆ’140q12βˆ’159q13βˆ’119q14βˆ’164q15βˆ’217q16βˆ’142q17βˆ’135q18βˆ’163q19βˆ’164q20βˆ’368q21βˆ’214q22βˆ’62q23βˆ’174q24+O(q25)11q - 3q^{2} - 11q^{3} + 11q^{4} - 27q^{5} - 22q^{6} - 32q^{7} - 15q^{8} + 14q^{9} - 98q^{10} - 103q^{11} - 140q^{12} - 159q^{13} - 119q^{14} - 164q^{15} - 217q^{16} - 142q^{17} - 135q^{18} - 163q^{19} - 164q^{20} - 368q^{21} - 214q^{22} - 62q^{23} - 174q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 11. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x11+14x10βˆ’85x9βˆ’1361x8+2076x7+34999x6βˆ’46085x5βˆ’269590x4+485437x3+176877x2βˆ’452668xβˆ’47687x^{11} + 14x^{10} - 85x^{9} - 1361x^{8} + 2076x^{7} + 34999x^{6} - 46085x^{5} - 269590x^{4} + 485437x^{3} + 176877x^{2} - 452668x - 47687

Atkin–Lehner signs

ppwpw_p
131βˆ’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’3βˆ’11βˆ’27βˆ’32βˆ’103βˆ’159βˆ’142βˆ’163βˆ’62βˆ’545βˆ’524βˆ’1011254βˆ’217496262βˆ’57βˆ’1827βˆ’747440βˆ’460βˆ’2148665βˆ’471βˆ’1182

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 0.11 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, 131]: [1, 34] (certified at ell = 2147482481)
  • T = [(2, 1), (3, 1)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(131, 4)[0]   # 131.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 131:4 1000