Atlas › Modular forms › Weight 8 › Level 9 › 9.8.a › 9.8.a.a

Newform orbit 9.8.a.a

$q$ -expansion

$q - 6q^{2} - 92q^{4} - 390q^{5} - 64q^{7} + 1320q^{8} + 2340q^{10} + 948q^{11} - 5098q^{13} + 384q^{14} + 3856q^{16} - 28386q^{17} - 8620q^{19} + 35880q^{20} - 5688q^{22} + 15288q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x + 396$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 9$ )

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
$\operatorname{tr} a_p$	−6	0	−390	−64	948	−5098	−28386	−8620	15288	−36510	−276808	268526	629718	685772	−583296	428058	−1306380	300662	−507244	−5560632	1369082	−6913720	4376748	8528310	−8826814

$\operatorname{tr} a_n$ 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 9, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 9]: [1, 3, 8] (certified at ell = 2147483647)
T = [(2, 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(9, 8)[0]   # 9.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 9:8 1000