🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 889 β€Ί 889.2.a β€Ί 889.2.a.c

Newform orbit 889.2.a.c

Trace form

The 16 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

16qβˆ’2q2βˆ’4q3+12q4βˆ’9q5βˆ’12q6βˆ’16q7βˆ’6q8+14q9βˆ’2q10βˆ’22q11βˆ’10q12βˆ’4q13+2q14βˆ’14q15+12q16βˆ’18q17βˆ’5q18βˆ’15q19βˆ’40q20+4q21βˆ’11q22βˆ’5q23βˆ’26q24+O(q25)16q - 2q^{2} - 4q^{3} + 12q^{4} - 9q^{5} - 12q^{6} - 16q^{7} - 6q^{8} + 14q^{9} - 2q^{10} - 22q^{11} - 10q^{12} - 4q^{13} + 2q^{14} - 14q^{15} + 12q^{16} - 18q^{17} - 5q^{18} - 15q^{19} - 40q^{20} + 4q^{21} - 11q^{22} - 5q^{23} - 26q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 16. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x16+6x15βˆ’23x14βˆ’165x13+218x12+1835x11βˆ’1263x10βˆ’10574x9+5684x8+32989x7βˆ’18945x6βˆ’51413x5+35124x4+29146x3βˆ’23456x2βˆ’300x+1681x^{16} + 6x^{15} - 23x^{14} - 165x^{13} + 218x^{12} + 1835x^{11} - 1263x^{10} - 10574x^{9} + 5684x^{8} + 32989x^{7} - 18945x^{6} - 51413x^{5} + 35124x^{4} + 29146x^{3} - 23456x^{2} - 300x + 1681

Atkin–Lehner signs

ppwpw_p
7+1+1
127+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=+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’2βˆ’4βˆ’9βˆ’16βˆ’22βˆ’4βˆ’18βˆ’15βˆ’5βˆ’12βˆ’32βˆ’2βˆ’45βˆ’3βˆ’4916βˆ’35βˆ’1117βˆ’81βˆ’15βˆ’34βˆ’39βˆ’32βˆ’4

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 0.258 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 label, dimension and tr⁑ap\operatorname{tr} a_p for every p<1000p < 1000 equal LMFDB's.

The engine's checks
  • dim M^+ at levels [1, 7, 127, 889]: [0, 1, 11, 86] (certified at ell = 2147483647)
  • T = [(2, 1), (3, 1)]; h of degree 63 = 1 x 63 (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(889, 2)[2]   # 889.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 889:2 1000