Atlas › Modular forms › Weight 6 › Level 9 › 9.6.a › 9.6.a.a

Newform orbit 9.6.a.a

$q$ -expansion

$q + 6q^{2} + 4q^{4} - 6q^{5} - 40q^{7} - 168q^{8} - 36q^{10} + 564q^{11} + 638q^{13} - 240q^{14} - 1136q^{16} - 882q^{17} - 556q^{19} - 24q^{20} + 3384q^{22} + 840q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{5}$ is a root of

$x + 12$

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	−6	−40	564	638	−882	−556	840	−4638	4400	−2410	6870	9644	18672	−33750	18084	39758	−23068	4248	−41110	21920	−82452	94086	49442

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.106 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, 6] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 1 = 1 x 1 (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, 6)[0]   # 9.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 9:6 1000