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

Newform orbit 182.4.a.h

Trace form

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

$2q + 4q^{2} + 2q^{3} + 8q^{4} + 4q^{5} + 4q^{6} + 14q^{7} + 16q^{8} + 34q^{9} + 8q^{10} + 34q^{11} + 8q^{12} - 26q^{13} + 28q^{14} - 82q^{15} + 32q^{16} + 150q^{17} + 68q^{18} + 236q^{19} + 16q^{20} + 14q^{21} + 68q^{22} + 2q^{23} + 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{43})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17}$ , whose minimal polynomial is

$x^{2} - 1234x + 336657$

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 even 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	2	4	14	34	−26	150	236	2	−14	−192	−166	−40	78	−628	−270	176	−36	328	−1602	−308	66	532	480	1028

$\operatorname{tr} a_n$ 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)[7]   # 182.4.a.h

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

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