Atlas › Modular forms › Weight 10 › Level 9 › 9.10.a › 9.10.a.a

Newform orbit 9.10.a.a

$q$ -expansion

$q - 18q^{2} - 188q^{4} + 1530q^{5} + 9128q^{7} + 12600q^{8} - 27540q^{10} - 21132q^{11} + 31214q^{13} - 164304q^{14} - 130544q^{16} + 279342q^{17} + 144020q^{19} - 287640q^{20} + 380376q^{22} + 1763496q^{23} + O(q^{25})$

Coefficient field

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

$x - 1512$

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
$\operatorname{tr} a_p$	−18	0	1530	9128	−21132	31214	279342	144020	1763496	−4692510	−369088	9347078	7226838	−23147476	−22971888	−78477174	20310660	−179339938	274528388	36342648	−247089526	191874800	276159276	678997350	−567657502

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

Sato–Tate

The normalized $a_p / 2p^{9/2}$ 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 0.29 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, 4, 10] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 2 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, 10)[0]   # 9.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 9:10 1000