Atlas › Modular forms › Weight 6 › Level 36 › 36.6.a › 36.6.a.a

Newform orbit 36.6.a.a

$q$ -expansion

$q - 54q^{5} - 88q^{7} - 540q^{11} - 418q^{13} - 594q^{17} + 836q^{19} + 4104q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + T_{7}$ is a root of

$x + 142$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 36$ )
3	— ( $3^2 \mid 36$ )

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
$\operatorname{tr} a_p$	0	0	−54	−88	−540	−418	−594	836	4104	594	4256	−298	−17226	−12100	1296	−19494	7668	−34738	21812	46872	67562	−76912	−67716	−29754	−122398

$\operatorname{tr} a_n$ 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 36, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.12 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, 4, 6, 9, 12, 18, 36]: [1, 2, 3, 4, 7, 6, 13, 17, 33] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 2 = 1 x 2 (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(36, 6)[0]   # 36.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 36:6 1000