🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 557 β€Ί 557.2.a β€Ί 557.2.a.c

Newform orbit 557.2.a.c

Trace form

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

18qβˆ’6q2βˆ’15q3+12q4βˆ’6q5βˆ’2q6βˆ’23q7βˆ’6q8+21q9βˆ’12q10βˆ’3q11βˆ’21q12βˆ’26q13+2q14βˆ’7q15βˆ’4q16βˆ’5q17βˆ’11q18βˆ’32q19βˆ’14q20+8q21βˆ’9q22βˆ’8q23+4q24+O(q25)18q - 6q^{2} - 15q^{3} + 12q^{4} - 6q^{5} - 2q^{6} - 23q^{7} - 6q^{8} + 21q^{9} - 12q^{10} - 3q^{11} - 21q^{12} - 26q^{13} + 2q^{14} - 7q^{15} - 4q^{16} - 5q^{17} - 11q^{18} - 32q^{19} - 14q^{20} + 8q^{21} - 9q^{22} - 8q^{23} + 4q^{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+21x17+161x16+433x15βˆ’857x14βˆ’7431x13βˆ’10117x12+23262x11+68245x10+3356x9βˆ’114160x8βˆ’39677x7+85091x6+22945x5βˆ’25517x4βˆ’4354x3+2316x2+271xβˆ’28x^{18} + 21x^{17} + 161x^{16} + 433x^{15} - 857x^{14} - 7431x^{13} - 10117x^{12} + 23262x^{11} + 68245x^{10} + 3356x^{9} - 114160x^{8} - 39677x^{7} + 85091x^{6} + 22945x^{5} - 25517x^{4} - 4354x^{3} + 2316x^{2} + 271x - 28

Atkin–Lehner signs

ppwpw_p
557+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’15βˆ’6βˆ’23βˆ’3βˆ’26βˆ’5βˆ’32βˆ’810βˆ’23βˆ’31βˆ’9βˆ’46βˆ’285βˆ’3110βˆ’78βˆ’19βˆ’3011βˆ’5119βˆ’73

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, 557]: [0, 47] (certified at ell = 2147475497)
  • T = [(2, 1), (3, 1)]; h of degree 46 = 1 x 46 (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(557, 2)[2]   # 557.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 557:2 1000