🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 487 β€Ί 487.2.a β€Ί 487.2.a.e

Newform orbit 487.2.a.e

Trace form

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

17qβˆ’8q2βˆ’6q3+16q4βˆ’19q5βˆ’8q6βˆ’11q7βˆ’21q8+9q9+q10βˆ’11q11βˆ’14q12βˆ’12q13βˆ’7q14βˆ’4q15+22q16βˆ’54q17βˆ’15q18βˆ’2q19βˆ’32q20βˆ’9q21βˆ’2q22βˆ’18q23βˆ’17q24+O(q25)17q - 8q^{2} - 6q^{3} + 16q^{4} - 19q^{5} - 8q^{6} - 11q^{7} - 21q^{8} + 9q^{9} + q^{10} - 11q^{11} - 14q^{12} - 12q^{13} - 7q^{14} - 4q^{15} + 22q^{16} - 54q^{17} - 15q^{18} - 2q^{19} - 32q^{20} - 9q^{21} - 2q^{22} - 18q^{23} - 17q^{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+14x16+51x15βˆ’100x14βˆ’888x13βˆ’603x12+4102x11+4468x10βˆ’8977x9βˆ’8253x8+10381x7+4364x6βˆ’4762x5βˆ’300x4+627x3βˆ’63x2βˆ’10x+1x^{17} + 14x^{16} + 51x^{15} - 100x^{14} - 888x^{13} - 603x^{12} + 4102x^{11} + 4468x^{10} - 8977x^{9} - 8253x^{8} + 10381x^{7} + 4364x^{6} - 4762x^{5} - 300x^{4} + 627x^{3} - 63x^{2} - 10x + 1

Atkin–Lehner signs

ppwpw_p
487+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’8βˆ’6βˆ’19βˆ’11βˆ’11βˆ’12βˆ’54βˆ’2βˆ’18βˆ’111βˆ’6βˆ’495βˆ’28βˆ’23βˆ’13βˆ’81717βˆ’4711βˆ’50βˆ’49βˆ’33

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 0.196 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, 487]: [0, 41] (certified at ell = 2147481883)
  • T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (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(487, 2)[4]   # 487.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 487:2 1000