Atlas › Modular forms › Weight 6 › Level 90 › 90.6.a › 90.6.a.g

Newform orbit 90.6.a.g

$q$ -expansion

$q + 4q^{2} + 16q^{4} + 25q^{5} + 164q^{7} + 64q^{8} + 100q^{10} - 720q^{11} + 698q^{13} + 656q^{14} + 256q^{16} + 2226q^{17} + 356q^{19} + 400q^{20} - 2880q^{22} + 1800q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{7} + T_{11}$ is a root of

$x + 556$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	— ( $3^2 \mid 90$ )
5	$-1$

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
$\operatorname{tr} a_p$	4	0	25	164	−720	698	2226	356	1800	−714	848	−11302	−9354	−5956	11160	−14106	−7920	−13450	−65476	−34560	86258	−108832	−10668	−10818	4418

$\operatorname{tr} a_n$ 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 90, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 5, 6, 9, 10, 15, 18, 30, 45, 90]: [1, 2, 3, 3, 7, 6, 9, 12, 17, 34, 32, 94] (certified at ell = 2147483629)
T = [(7, 1), (11, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 3 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(90, 6)[6]   # 90.6.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 90:6 1000