Atlas › Modular forms › Weight 4 › Level 182 › 182.4.a › 182.4.a.e

Newform orbit 182.4.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} - 10q^{3} + 8q^{4} - 8q^{5} + 20q^{6} + 14q^{7} - 16q^{8} + 50q^{9} + 16q^{10} + 46q^{11} - 40q^{12} - 26q^{13} - 28q^{14} + 58q^{15} + 32q^{16} + 18q^{17} - 100q^{18} - 128q^{19} - 32q^{20} - 70q^{21} - 92q^{22} + 42q^{23} + 80q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17}$ , whose minimal polynomial is

$x^{2} - 322x - 84671$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$-1$
13	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−4	−10	−8	14	46	−26	18	−128	42	90	−100	−418	−624	−850	536	−166	608	−132	752	42	−200	−262	−448	900	−3136

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 7, 13, 14, 26, 91, 182]: [1, 2, 3, 5, 8, 13, 30, 88] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (17, 7)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 7 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(182, 4)[4]   # 182.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 182:4 1000