🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 683 β€Ί 683.2.a β€Ί 683.2.a.b

Newform orbit 683.2.a.b

Trace form

The 24 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

24qβˆ’4q2βˆ’8q3+18q4βˆ’16q5βˆ’14q6βˆ’24q7βˆ’12q8+18q9βˆ’19q10βˆ’4q11βˆ’24q12βˆ’56q13βˆ’7q15+10q16βˆ’21q17βˆ’9q18βˆ’23q19βˆ’18q20βˆ’20q21βˆ’31q22βˆ’29q24+O(q25)24q - 4q^{2} - 8q^{3} + 18q^{4} - 16q^{5} - 14q^{6} - 24q^{7} - 12q^{8} + 18q^{9} - 19q^{10} - 4q^{11} - 24q^{12} - 56q^{13} - 7q^{15} + 10q^{16} - 21q^{17} - 9q^{18} - 23q^{19} - 18q^{20} - 20q^{21} - 31q^{22} - 29q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 24. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x24+12x23+8x22βˆ’387x21βˆ’1034x20+4631x19+16648x18βˆ’29536x17βˆ’121094x16+127948x15+469333x14βˆ’436933x13βˆ’958573x12+1043858x11+831319x10βˆ’1294168x9βˆ’2406x8+525520x7βˆ’132235x6βˆ’70270x5+26607x4+1787x3βˆ’1094x2+4x+11x^{24} + 12x^{23} + 8x^{22} - 387x^{21} - 1034x^{20} + 4631x^{19} + 16648x^{18} - 29536x^{17} - 121094x^{16} + 127948x^{15} + 469333x^{14} - 436933x^{13} - 958573x^{12} + 1043858x^{11} + 831319x^{10} - 1294168x^{9} - 2406x^{8} + 525520x^{7} - 132235x^{6} - 70270x^{5} + 26607x^{4} + 1787x^{3} - 1094x^{2} + 4x + 11

Atkin–Lehner signs

ppwpw_p
683+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’4βˆ’8βˆ’16βˆ’24βˆ’4βˆ’56βˆ’21βˆ’230βˆ’16βˆ’14βˆ’71βˆ’4βˆ’27βˆ’17βˆ’2118βˆ’75βˆ’1534βˆ’91βˆ’3βˆ’158βˆ’80

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 0.23 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, 683]: [0, 58] (certified at ell = 2147483647)
  • T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(683, 2)[1]   # 683.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 683:2 1000