Atlas › Modular forms › Weight 8 › Level 16 › 16.8.a › 16.8.a.a

Newform orbit 16.8.a.a

$q$ -expansion

$q - 44q^{3} + 430q^{5} + 1224q^{7} - 251q^{9} + 3164q^{11} + 6118q^{13} - 18920q^{15} - 16270q^{17} + 5476q^{19} - 53856q^{21} - 1576q^{23} + O(q^{25})$

Coefficient field

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

$x - 386$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 16$ )

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	−44	430	1224	3164	6118	−16270	5476	−1576	122838	−251360	−52338	−319398	−710788	−284112	296062	897548	−884810	−4659692	2710792	−5670854	5124176	1563556	11605674	10931618

$\operatorname{tr} a_n$ 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 16, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 8, 16]: [1, 3, 5, 9, 16] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(16, 8)[0]   # 16.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 16:8 1000