🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 273 β€Ί 273.2.a β€Ί 273.2.a.d

Newform orbit 273.2.a.d

Trace form

The 3 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

3qβˆ’2q2βˆ’3q3+4q4βˆ’3q5+2q6βˆ’3q7βˆ’12q8+3q9βˆ’2q10βˆ’2q11βˆ’4q12βˆ’3q13+2q14+3q15+18q16βˆ’8q17βˆ’2q18βˆ’7q19βˆ’10q20+3q21βˆ’8q22βˆ’9q23+12q24+O(q25)3q - 2q^{2} - 3q^{3} + 4q^{4} - 3q^{5} + 2q^{6} - 3q^{7} - 12q^{8} + 3q^{9} - 2q^{10} - 2q^{11} - 4q^{12} - 3q^{13} + 2q^{14} + 3q^{15} + 18q^{16} - 8q^{17} - 2q^{18} - 7q^{19} - 10q^{20} + 3q^{21} - 8q^{22} - 9q^{23} + 12q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 3. It is generated by the eigenvalue Ξ±\alpha of T=T2+3T5T = T_{2} + 3T_{5}, whose minimal polynomial is

x3+11x2βˆ’15xβˆ’293x^{3} + 11x^{2} - 15x - 293

Atkin–Lehner signs

ppwpw_p
3+1+1
7+1+1
13+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’2βˆ’3βˆ’3βˆ’3βˆ’2βˆ’3βˆ’8βˆ’7βˆ’9βˆ’1βˆ’712βˆ’10βˆ’1βˆ’17βˆ’5βˆ’12102βˆ’4βˆ’5βˆ’13βˆ’1βˆ’13βˆ’9

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 0.198 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, 7, 13, 21, 39, 91, 273]: [0, 1, 1, 1, 4, 6, 10, 40] (certified at ell = 2147483629)
  • T = [(2, 1), (5, 3)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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(273, 2)[3]   # 273.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 273:2 1000