🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 965 β€Ί 965.2.a β€Ί 965.2.a.c

Newform orbit 965.2.a.c

Trace form

The 8 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

8qβˆ’2q2βˆ’6q3βˆ’2q4+8q5+4q6βˆ’3q7βˆ’6q8βˆ’2q9βˆ’2q10βˆ’14q11βˆ’q12βˆ’6q15βˆ’6q16+6q17βˆ’5q18βˆ’15q19βˆ’2q20βˆ’13q21+6q22βˆ’5q23+7q24+O(q25)8q - 2q^{2} - 6q^{3} - 2q^{4} + 8q^{5} + 4q^{6} - 3q^{7} - 6q^{8} - 2q^{9} - 2q^{10} - 14q^{11} - q^{12} - 6q^{15} - 6q^{16} + 6q^{17} - 5q^{18} - 15q^{19} - 2q^{20} - 13q^{21} + 6q^{22} - 5q^{23} + 7q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 8. It is generated by the eigenvalue Ξ±\alpha of T=T2+3T3T = T_{2} + 3T_{3}, whose minimal polynomial is

x8+20x7+82x6βˆ’533x5βˆ’3432x4+2807x3+26848x2βˆ’1943xβˆ’32519x^{8} + 20x^{7} + 82x^{6} - 533x^{5} - 3432x^{4} + 2807x^{3} + 26848x^{2} - 1943x - 32519

Atkin–Lehner signs

ppwpw_p
5βˆ’1-1
193βˆ’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’68βˆ’3βˆ’1406βˆ’15βˆ’5βˆ’13βˆ’29βˆ’3βˆ’3βˆ’391βˆ’3βˆ’13βˆ’18βˆ’22βˆ’26βˆ’119βˆ’30βˆ’3313

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 0.302 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, 5, 193, 965]: [0, 1, 16, 98] (certified at ell = 2147483137)
  • T = [(2, 1), (3, 3)]; h of degree 65 = 1 x 65 (phi(ord eps) x dim S_k^new); 9 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(965, 2)[2]   # 965.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 965:2 1000