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Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 677 β€Ί 677.2.a β€Ί 677.2.a.c

Newform orbit 677.2.a.c

Trace form

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

18qβˆ’2q2βˆ’12q3+10q4βˆ’3q5βˆ’6q6βˆ’18q7βˆ’9q8+14q9βˆ’17q10βˆ’11q11βˆ’16q12βˆ’19q13βˆ’8q14βˆ’20q15βˆ’6q16βˆ’11q17βˆ’2q18βˆ’15q19+q20βˆ’3q21βˆ’21q22βˆ’14q23+7q24+O(q25)18q - 2q^{2} - 12q^{3} + 10q^{4} - 3q^{5} - 6q^{6} - 18q^{7} - 9q^{8} + 14q^{9} - 17q^{10} - 11q^{11} - 16q^{12} - 19q^{13} - 8q^{14} - 20q^{15} - 6q^{16} - 11q^{17} - 2q^{18} - 15q^{19} + q^{20} - 3q^{21} - 21q^{22} - 14q^{23} + 7q^{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+14x17+47x16βˆ’162x15βˆ’1140x14βˆ’122x13+9009x12+8290x11βˆ’34993x10βˆ’41573x9+75500x8+87692x7βˆ’93529x6βˆ’82854x5+63196x4+27921x3βˆ’18687x2+213x+26x^{18} + 14x^{17} + 47x^{16} - 162x^{15} - 1140x^{14} - 122x^{13} + 9009x^{12} + 8290x^{11} - 34993x^{10} - 41573x^{9} + 75500x^{8} + 87692x^{7} - 93529x^{6} - 82854x^{5} + 63196x^{4} + 27921x^{3} - 18687x^{2} + 213x + 26

Atkin–Lehner signs

ppwpw_p
677+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’2βˆ’12βˆ’3βˆ’18βˆ’11βˆ’19βˆ’11βˆ’15βˆ’14βˆ’5βˆ’7βˆ’4311βˆ’67βˆ’18βˆ’97βˆ’16βˆ’65βˆ’6βˆ’44βˆ’8βˆ’4βˆ’1βˆ’19

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 0.233 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, 677]: [0, 57] (certified at ell = 2147480297)
  • T = [(2, 1), (3, 1)]; h of degree 56 = 1 x 56 (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(677, 2)[2]   # 677.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 677:2 1000