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

Newform orbit 39.8.a.d

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 135q^{3} + 454q^{4} + 774q^{5} + 1420q^{7} + 1170q^{8} + 3645q^{9} - 5672q^{10} - 2280q^{11} + 12258q^{12} + 10985q^{13} - 18672q^{14} + 20898q^{15} + 71410q^{16} + 54330q^{17} + 83832q^{19} + 123876q^{20} + 38340q^{21} + 39896q^{22} + 2736q^{23} + 31590q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{5} - 774x^{4} + 5793x^{3} + 103763376x^{2} - 22000980456x + 1017490921920$

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 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
$\operatorname{tr} a_p$	0	135	774	1420	−2280	10985	54330	83832	2736	80622	18332	298166	−720030	−1159956	−180780	−1903386	2455752	1361078	7533704	2672820	10580722	4006864	359208	5879106	8253386

$\operatorname{tr} a_n$ 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)[3]   # 39.8.a.d

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

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