Atlas › Modular forms › Weight 8 › Level 39 › 39.8.a › 39.8.a.a

Newform orbit 39.8.a.a

Trace form

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

$2q - 8q^{2} + 54q^{3} + 8q^{4} - 132q^{5} - 216q^{6} - 1116q^{7} - 864q^{8} + 1458q^{9} + 760q^{10} - 292q^{11} + 216q^{12} - 4394q^{13} - 1336q^{14} - 3564q^{15} - 18912q^{16} - 16004q^{17} - 5832q^{18} - 34836q^{19} - 2384q^{20} - 30132q^{21} - 89312q^{22} - 84296q^{23} - 23328q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{29})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 140x + 4436$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
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
$\operatorname{tr} a_p$	−8	54	−132	−1116	−292	−4394	−16004	−34836	−84296	−107132	9820	−17020	542092	−260512	1167660	−22804	−314036	−1240028	−3256228	−227516	1100484	−12543984	−2193436	−4415748	4898660

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.281 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, 3, 13, 39]: [1, 3, 9, 34] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 5 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(39, 8)[0]   # 39.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 39:8 1000