🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 601 β€Ί 601.2.a β€Ί 601.2.a.a

Newform orbit 601.2.a.a

Trace form

The 20 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

20qβˆ’5q2βˆ’7q3+11q4βˆ’12q5βˆ’8q6βˆ’11q7βˆ’12q8+3q9βˆ’7q10βˆ’34q11βˆ’16q12βˆ’3q13βˆ’26q14βˆ’20q15+q16βˆ’13q17+2q18βˆ’14q19βˆ’15q20βˆ’4q21+7q22βˆ’47q23βˆ’3q24+O(q25)20q - 5q^{2} - 7q^{3} + 11q^{4} - 12q^{5} - 8q^{6} - 11q^{7} - 12q^{8} + 3q^{9} - 7q^{10} - 34q^{11} - 16q^{12} - 3q^{13} - 26q^{14} - 20q^{15} + q^{16} - 13q^{17} + 2q^{18} - 14q^{19} - 15q^{20} - 4q^{21} + 7q^{22} - 47q^{23} - 3q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 20. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x20+12x19+23x18βˆ’227x17βˆ’889x16+1179x15+8477x14βˆ’10x13βˆ’35522x12βˆ’13615x11+76003x10+35031x9βˆ’89376x8βˆ’38312x7+59500x6+20887x5βˆ’22150x4βˆ’5506x3+4277x2+549xβˆ’337x^{20} + 12x^{19} + 23x^{18} - 227x^{17} - 889x^{16} + 1179x^{15} + 8477x^{14} - 10x^{13} - 35522x^{12} - 13615x^{11} + 76003x^{10} + 35031x^{9} - 89376x^{8} - 38312x^{7} + 59500x^{6} + 20887x^{5} - 22150x^{4} - 5506x^{3} + 4277x^{2} + 549x - 337

Atkin–Lehner signs

ppwpw_p
601+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’5βˆ’7βˆ’12βˆ’11βˆ’34βˆ’3βˆ’13βˆ’14βˆ’47βˆ’29βˆ’261βˆ’292βˆ’37βˆ’22βˆ’21βˆ’1418βˆ’884βˆ’48βˆ’1βˆ’145

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 0.208 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, 601]: [0, 50] (certified at ell = 2147482801)
  • T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (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(601, 2)[0]   # 601.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 601:2 1000