🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 401 β€Ί 401.2.a β€Ί 401.2.a.a

Newform orbit 401.2.a.a

Trace form

The 12 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

12qβˆ’3q2βˆ’5q3+5q4βˆ’7q5βˆ’9q6βˆ’20q7βˆ’3q8+3q9βˆ’11q10βˆ’11q11βˆ’9q12βˆ’11q13βˆ’3q14βˆ’11q15βˆ’9q16+q17βˆ’q18βˆ’34q19βˆ’5q20βˆ’3q21+3q22βˆ’7q23βˆ’9q24+O(q25)12q - 3q^{2} - 5q^{3} + 5q^{4} - 7q^{5} - 9q^{6} - 20q^{7} - 3q^{8} + 3q^{9} - 11q^{10} - 11q^{11} - 9q^{12} - 11q^{13} - 3q^{14} - 11q^{15} - 9q^{16} + q^{17} - q^{18} - 34q^{19} - 5q^{20} - 3q^{21} + 3q^{22} - 7q^{23} - 9q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 12. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x12+8x11+7x10βˆ’70x9βˆ’117x8+161x7+318x6βˆ’114x5βˆ’274x4+27x3+78x2βˆ’2xβˆ’7x^{12} + 8x^{11} + 7x^{10} - 70x^{9} - 117x^{8} + 161x^{7} + 318x^{6} - 114x^{5} - 274x^{4} + 27x^{3} + 78x^{2} - 2x - 7

Atkin–Lehner signs

ppwpw_p
401+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’3βˆ’5βˆ’7βˆ’20βˆ’11βˆ’111βˆ’34βˆ’7βˆ’6βˆ’523βˆ’16βˆ’2βˆ’319βˆ’1βˆ’246βˆ’15βˆ’20βˆ’5317βˆ’112

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 0.181 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, 401]: [0, 34] (certified at ell = 2147482801)
  • T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 4 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(401, 2)[0]   # 401.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 401:2 1000