🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 813 β€Ί 813.2.a β€Ί 813.2.a.e

Newform orbit 813.2.a.e

Trace form

The 15 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

15qβˆ’4q2βˆ’15q3+18q4βˆ’10q5+4q6βˆ’9q7βˆ’12q8+15q9βˆ’6q10βˆ’19q11βˆ’18q12βˆ’5q13βˆ’15q14+10q15+16q16βˆ’16q17βˆ’4q18βˆ’7q19βˆ’41q20+9q21βˆ’14q22βˆ’21q23+12q24+O(q25)15q - 4q^{2} - 15q^{3} + 18q^{4} - 10q^{5} + 4q^{6} - 9q^{7} - 12q^{8} + 15q^{9} - 6q^{10} - 19q^{11} - 18q^{12} - 5q^{13} - 15q^{14} + 10q^{15} + 16q^{16} - 16q^{17} - 4q^{18} - 7q^{19} - 41q^{20} + 9q^{21} - 14q^{22} - 21q^{23} + 12q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 15. It is generated by the eigenvalue Ξ±\alpha of T=T2+T5T = T_{2} + T_{5}, whose minimal polynomial is

x15+14x14+27x13βˆ’386x12βˆ’1490x11+3434x10+18807x9βˆ’11895x8βˆ’102285x7+9096x6+270146x5+28853x4βˆ’332788x3βˆ’36678x2+145279xβˆ’15992x^{15} + 14x^{14} + 27x^{13} - 386x^{12} - 1490x^{11} + 3434x^{10} + 18807x^{9} - 11895x^{8} - 102285x^{7} + 9096x^{6} + 270146x^{5} + 28853x^{4} - 332788x^{3} - 36678x^{2} + 145279x - 15992

Atkin–Lehner signs

ppwpw_p
3+1+1
271+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’4βˆ’15βˆ’10βˆ’9βˆ’19βˆ’5βˆ’16βˆ’7βˆ’21βˆ’8βˆ’7βˆ’5βˆ’18βˆ’21βˆ’44βˆ’29βˆ’45βˆ’4βˆ’40βˆ’26βˆ’142βˆ’77βˆ’26βˆ’10

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 0.246 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, 3, 271, 813]: [0, 1, 23, 92] (certified at ell = 2147482801)
  • T = [(2, 1), (5, 1)]; h of degree 45 = 1 x 45 (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(813, 2)[4]   # 813.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 813:2 1000