🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 619 β€Ί 619.2.a β€Ί 619.2.a.a

Newform orbit 619.2.a.a

Trace form

The 21 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

21qβˆ’9q2βˆ’5q3+15q4βˆ’21q5βˆ’6q6βˆ’4q7βˆ’21q8+6q9+q10βˆ’27q11βˆ’8q12βˆ’11q13βˆ’19q14βˆ’10q15+11q16βˆ’14q17βˆ’14q18βˆ’15q19βˆ’25q20βˆ’42q21+12q22βˆ’14q23βˆ’8q24+O(q25)21q - 9q^{2} - 5q^{3} + 15q^{4} - 21q^{5} - 6q^{6} - 4q^{7} - 21q^{8} + 6q^{9} + q^{10} - 27q^{11} - 8q^{12} - 11q^{13} - 19q^{14} - 10q^{15} + 11q^{16} - 14q^{17} - 14q^{18} - 15q^{19} - 25q^{20} - 42q^{21} + 12q^{22} - 14q^{23} - 8q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 21. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x21+14x20+41x19βˆ’251x18βˆ’1477x17+654x16+15951x15+11966x14βˆ’82845x13βˆ’105277x12+231127x11+366363x10βˆ’346120x9βˆ’642344x8+252294x7+568419x6βˆ’67086x5βˆ’229352x4+1596x3+35853x2+558xβˆ’1089x^{21} + 14x^{20} + 41x^{19} - 251x^{18} - 1477x^{17} + 654x^{16} + 15951x^{15} + 11966x^{14} - 82845x^{13} - 105277x^{12} + 231127x^{11} + 366363x^{10} - 346120x^{9} - 642344x^{8} + 252294x^{7} + 568419x^{6} - 67086x^{5} - 229352x^{4} + 1596x^{3} + 35853x^{2} + 558x - 1089

Atkin–Lehner signs

ppwpw_p
619+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’9βˆ’5βˆ’21βˆ’4βˆ’27βˆ’11βˆ’14βˆ’15βˆ’14βˆ’78βˆ’8βˆ’23βˆ’592βˆ’12βˆ’36βˆ’17βˆ’2215βˆ’56βˆ’25βˆ’1βˆ’1221

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 0.218 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, 619]: [0, 52] (certified at ell = 2147482021)
  • T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (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(619, 2)[0]   # 619.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 619:2 1000