Atlas › Modular forms › Weight 8 › Level 18 › 18.8.a › 18.8.a.a

Newform orbit 18.8.a.a

$q$ -expansion

$q - 8q^{2} + 64q^{4} + 114q^{5} - 1576q^{7} - 512q^{8} - 912q^{10} - 7332q^{11} - 3802q^{13} + 12608q^{14} + 4096q^{16} + 6606q^{17} + 24860q^{19} + 7296q^{20} + 58656q^{22} - 41448q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + T_{7}$ is a root of

$x + 1462$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	— ( $3^2 \mid 18$ )

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
$\operatorname{tr} a_p$	−8	0	114	−1576	−7332	−3802	6606	24860	−41448	41610	33152	−36466	639078	−156412	433776	−786078	−745140	−1660618	−3290836	−5716152	2659898	3807440	−2229468	−5991210	−4060126

$\operatorname{tr} a_n$ 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 18, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.254 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, 3, 6, 9, 18]: [1, 3, 3, 9, 8, 23] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 2 = 1 x 2 (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(18, 8)[0]   # 18.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 18:8 1000