Atlas › Modular forms › Weight 8 › Level 56 › 56.8.a › 56.8.a.a

Newform orbit 56.8.a.a

$q$ -expansion

$q - 18q^{3} + 160q^{5} - 343q^{7} - 1863q^{9} + 5704q^{11} + 1388q^{13} - 2880q^{15} - 31434q^{17} - 19966q^{19} + 6174q^{21} - 77136q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x - 142$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 56$ )
7	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	−18	160	−343	5704	1388	−31434	−19966	−77136	−193374	−26356	204346	−663050	−335920	1119812	112782	536154	−1170264	3890660	2505344	−1435070	176536	−6211622	−4729062	−2129562

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ for the primes $p < 1000$ not dividing 56, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.313 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, 4, 7, 8, 14, 28, 56]: [1, 3, 5, 5, 9, 16, 31, 60] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 4 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(56, 8)[0]   # 56.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 56:8 1000