🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 697 β€Ί 697.2.a β€Ί 697.2.a.e

Newform orbit 697.2.a.e

Trace form

The 15 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

15qβˆ’5q2βˆ’8q3+17q4βˆ’9q5βˆ’8q6βˆ’19q7βˆ’15q8+15q9+3q10βˆ’14q11βˆ’13q12βˆ’11q13+7q14βˆ’2q15+17q16βˆ’15q17βˆ’24q18+q19βˆ’40q20+4q21βˆ’10q22βˆ’30q23βˆ’19q24+O(q25)15q - 5q^{2} - 8q^{3} + 17q^{4} - 9q^{5} - 8q^{6} - 19q^{7} - 15q^{8} + 15q^{9} + 3q^{10} - 14q^{11} - 13q^{12} - 11q^{13} + 7q^{14} - 2q^{15} + 17q^{16} - 15q^{17} - 24q^{18} + q^{19} - 40q^{20} + 4q^{21} - 10q^{22} - 30q^{23} - 19q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 15. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x15+13x14+39x13βˆ’125x12βˆ’689x11+228x10+3791x9+515x8βˆ’8439x7βˆ’905x6+5673x5+985x4βˆ’997x3βˆ’140x2+41x+5x^{15} + 13x^{14} + 39x^{13} - 125x^{12} - 689x^{11} + 228x^{10} + 3791x^{9} + 515x^{8} - 8439x^{7} - 905x^{6} + 5673x^{5} + 985x^{4} - 997x^{3} - 140x^{2} + 41x + 5

Atkin–Lehner signs

ppwpw_p
17+1+1
41+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’5βˆ’8βˆ’9βˆ’19βˆ’14βˆ’11βˆ’151βˆ’30βˆ’20βˆ’14βˆ’5βˆ’154βˆ’626βˆ’7βˆ’87βˆ’36βˆ’6βˆ’28βˆ’3312βˆ’35

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 0.225 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, 17, 41, 697]: [0, 2, 4, 64] (certified at ell = 2147482801)
  • T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (phi(ord eps) x dim S_k^new); 6 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(697, 2)[4]   # 697.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 697:2 1000