🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 191 β€Ί 191.4.a β€Ί 191.4.a.a

Newform orbit 191.4.a.a

Trace form

The 17 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

17qβˆ’4q2βˆ’11q3+34q4βˆ’21q5βˆ’57q6βˆ’38q7βˆ’45q8+26q9βˆ’146q10βˆ’54q11βˆ’93q12βˆ’261q13βˆ’84q14βˆ’162q15βˆ’10q16βˆ’305q17βˆ’79q18βˆ’400q19βˆ’303q20βˆ’402q21βˆ’149q22βˆ’179q23βˆ’534q24+O(q25)17q - 4q^{2} - 11q^{3} + 34q^{4} - 21q^{5} - 57q^{6} - 38q^{7} - 45q^{8} + 26q^{9} - 146q^{10} - 54q^{11} - 93q^{12} - 261q^{13} - 84q^{14} - 162q^{15} - 10q^{16} - 305q^{17} - 79q^{18} - 400q^{19} - 303q^{20} - 402q^{21} - 149q^{22} - 179q^{23} - 534q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 17. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x17+15x16βˆ’158x15βˆ’2710x14+9519x13+179446x12βˆ’317075x11βˆ’5319753x10+6391752x9+67448214x8βˆ’46023009x7βˆ’355056244x6+4763082x5+633857387x4+466098455x3+60334400x2βˆ’30721035xβˆ’6717759x^{17} + 15x^{16} - 158x^{15} - 2710x^{14} + 9519x^{13} + 179446x^{12} - 317075x^{11} - 5319753x^{10} + 6391752x^{9} + 67448214x^{8} - 46023009x^{7} - 355056244x^{6} + 4763082x^{5} + 633857387x^{4} + 466098455x^{3} + 60334400x^{2} - 30721035x - 6717759

Atkin–Lehner signs

ppwpw_p
191βˆ’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’4βˆ’11βˆ’21βˆ’38βˆ’54βˆ’261βˆ’305βˆ’400βˆ’179βˆ’432βˆ’646βˆ’430βˆ’1084βˆ’785βˆ’612536593βˆ’2058137598βˆ’1834βˆ’12111146βˆ’1760βˆ’4839

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 0.134 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, 191]: [1, 49] (certified at ell = 2147483171)
  • T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 7 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(191, 4)[0]   # 191.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 191:4 1000