🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 613 β€Ί 613.2.a β€Ί 613.2.a.b

Newform orbit 613.2.a.b

Trace form

The 18 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

18qβˆ’6q2βˆ’13q3+12q4βˆ’16q5βˆ’2q6βˆ’8q7βˆ’18q8+31q9+7q10βˆ’23q11βˆ’18q12βˆ’21q13+5q14βˆ’2q15+8q16βˆ’14q17βˆ’30q18βˆ’11q19βˆ’31q20βˆ’2q21+20q22βˆ’22q23+5q24+O(q25)18q - 6q^{2} - 13q^{3} + 12q^{4} - 16q^{5} - 2q^{6} - 8q^{7} - 18q^{8} + 31q^{9} + 7q^{10} - 23q^{11} - 18q^{12} - 21q^{13} + 5q^{14} - 2q^{15} + 8q^{16} - 14q^{17} - 30q^{18} - 11q^{19} - 31q^{20} - 2q^{21} + 20q^{22} - 22q^{23} + 5q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 18. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x18+19x17+116x16+76x15βˆ’1774x14βˆ’5085x13+5847x12+34758x11+3433x10βˆ’89357x9βˆ’26062x8+104837x7+19326x6βˆ’53732x5βˆ’4104x4+10702x3+676x2βˆ’711xβˆ’81x^{18} + 19x^{17} + 116x^{16} + 76x^{15} - 1774x^{14} - 5085x^{13} + 5847x^{12} + 34758x^{11} + 3433x^{10} - 89357x^{9} - 26062x^{8} + 104837x^{7} + 19326x^{6} - 53732x^{5} - 4104x^{4} + 10702x^{3} + 676x^{2} - 711x - 81

Atkin–Lehner signs

ppwpw_p
613+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’6βˆ’13βˆ’16βˆ’8βˆ’23βˆ’21βˆ’14βˆ’11βˆ’2213βˆ’4βˆ’22βˆ’3914βˆ’45βˆ’15βˆ’4616βˆ’28βˆ’16177βˆ’58βˆ’18βˆ’8

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 0.212 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, 613]: [0, 51] (certified at ell = 2147478013)
  • T = [(2, 1), (3, 1)]; h of degree 50 = 1 x 50 (phi(ord eps) x dim S_k^new); 5 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(613, 2)[1]   # 613.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 613:2 1000