🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 541 β€Ί 541.2.a β€Ί 541.2.a.a

Newform orbit 541.2.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’12q3+15q4βˆ’17q5βˆ’11q6βˆ’6q7βˆ’15q8+14q9βˆ’4q10βˆ’40q11βˆ’17q12βˆ’5q13βˆ’20q14βˆ’8q15+13q16βˆ’10q17+q18βˆ’28q19βˆ’39q20βˆ’14q21+6q22βˆ’17q23βˆ’20q24+O(q25)20q - 5q^{2} - 12q^{3} + 15q^{4} - 17q^{5} - 11q^{6} - 6q^{7} - 15q^{8} + 14q^{9} - 4q^{10} - 40q^{11} - 17q^{12} - 5q^{13} - 20q^{14} - 8q^{15} + 13q^{16} - 10q^{17} + q^{18} - 28q^{19} - 39q^{20} - 14q^{21} + 6q^{22} - 17q^{23} - 20q^{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+17x19+91x18+12x17βˆ’1441x16βˆ’3371x15+6564x14+29408x13βˆ’2280x12βˆ’104894x11βˆ’58511x10+170965x9+159486x8βˆ’104570x7βˆ’156707x6βˆ’9880x5+51031x4+22880x3+1971x2βˆ’668xβˆ’113x^{20} + 17x^{19} + 91x^{18} + 12x^{17} - 1441x^{16} - 3371x^{15} + 6564x^{14} + 29408x^{13} - 2280x^{12} - 104894x^{11} - 58511x^{10} + 170965x^{9} + 159486x^{8} - 104570x^{7} - 156707x^{6} - 9880x^{5} + 51031x^{4} + 22880x^{3} + 1971x^{2} - 668x - 113

Atkin–Lehner signs

ppwpw_p
541+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’12βˆ’17βˆ’6βˆ’40βˆ’5βˆ’10βˆ’28βˆ’17βˆ’25βˆ’31βˆ’11βˆ’35βˆ’6βˆ’296βˆ’110βˆ’14βˆ’3βˆ’276βˆ’8βˆ’60βˆ’6021

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 0.204 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, 541]: [0, 45] (certified at ell = 2147482801)
  • T = [(2, 1), (3, 1)]; h of degree 44 = 1 x 44 (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(541, 2)[0]   # 541.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 541:2 1000