🌿 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.a

Newform orbit 889.2.a.a

Trace form

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

12qβˆ’7q2βˆ’4q3+9q4βˆ’7q5βˆ’2q6+12q7βˆ’15q8βˆ’2q9βˆ’2q10βˆ’22q11+4q12βˆ’7q14βˆ’10q15+7q16βˆ’6q17βˆ’10q18βˆ’7q19βˆ’8q20βˆ’4q21+13q22βˆ’29q23βˆ’6q24+O(q25)12q - 7q^{2} - 4q^{3} + 9q^{4} - 7q^{5} - 2q^{6} + 12q^{7} - 15q^{8} - 2q^{9} - 2q^{10} - 22q^{11} + 4q^{12} - 7q^{14} - 10q^{15} + 7q^{16} - 6q^{17} - 10q^{18} - 7q^{19} - 8q^{20} - 4q^{21} + 13q^{22} - 29q^{23} - 6q^{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+11x11+29x10βˆ’64x9βˆ’373x8βˆ’298x7+611x6+898x5+58x4βˆ’342x3βˆ’146x2βˆ’21xβˆ’1x^{12} + 11x^{11} + 29x^{10} - 64x^{9} - 373x^{8} - 298x^{7} + 611x^{6} + 898x^{5} + 58x^{4} - 342x^{3} - 146x^{2} - 21x - 1

Atkin–Lehner signs

ppwpw_p
7βˆ’1-1
127βˆ’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’7βˆ’4βˆ’712βˆ’220βˆ’6βˆ’7βˆ’29βˆ’22βˆ’16βˆ’4βˆ’2111βˆ’31βˆ’38βˆ’15βˆ’3βˆ’1βˆ’57βˆ’7βˆ’18βˆ’2164

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)[0]   # 889.2.a.a

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

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