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

Newform orbit 90.6.a.d

$q$ -expansion

$q + 4q^{2} + 16q^{4} - 25q^{5} - 172q^{7} + 64q^{8} - 100q^{10} - 132q^{11} - 946q^{13} - 688q^{14} + 256q^{16} + 222q^{17} + 500q^{19} - 400q^{20} - 528q^{22} - 3564q^{23} + O(q^{25})$

Coefficient field

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

$x + 304$

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	−172	−132	−946	222	500	−3564	−2190	2312	−11242	−1242	20624	−6588	21066	−7980	16622	1808	24528	20474	−46240	51576	110310	−78382

$\operatorname{tr} a_n$ 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)[3]   # 90.6.a.d

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

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