🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 803 β€Ί 803.2.a β€Ί 803.2.a.c

Newform orbit 803.2.a.c

Trace form

The 9 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

9q+3q2βˆ’5q3+5q4βˆ’q5βˆ’3q6βˆ’q7βˆ’6q8βˆ’2q9βˆ’16q10βˆ’9q11βˆ’8q12βˆ’25q13βˆ’16q14βˆ’10q15βˆ’11q16+8q18βˆ’q19+3q20βˆ’7q21βˆ’3q22βˆ’4q23βˆ’6q24+O(q25)9q + 3q^{2} - 5q^{3} + 5q^{4} - q^{5} - 3q^{6} - q^{7} - 6q^{8} - 2q^{9} - 16q^{10} - 9q^{11} - 8q^{12} - 25q^{13} - 16q^{14} - 10q^{15} - 11q^{16} + 8q^{18} - q^{19} + 3q^{20} - 7q^{21} - 3q^{22} - 4q^{23} - 6q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 9. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x9+2x8βˆ’19x7βˆ’27x6+105x5+46x4βˆ’194x3+107x2βˆ’20x+1x^{9} + 2x^{8} - 19x^{7} - 27x^{6} + 105x^{5} + 46x^{4} - 194x^{3} + 107x^{2} - 20x + 1

Atkin–Lehner signs

ppwpw_p
11+1+1
73+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_p3βˆ’5βˆ’1βˆ’1βˆ’9βˆ’250βˆ’1βˆ’4βˆ’13βˆ’6βˆ’28110βˆ’8βˆ’25βˆ’3βˆ’6βˆ’43βˆ’21βˆ’91926βˆ’274

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 0.24 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, 11, 73, 803]: [0, 2, 6, 76] (certified at ell = 2147482801)
  • T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (phi(ord eps) x dim S_k^new); 6 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(803, 2)[2]   # 803.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 803:2 1000