Atlas › Modular forms › Weight 6 › Level 16 › 16.6.a › 16.6.a.b

Newform orbit 16.6.a.b

$q$ -expansion

$q + 12q^{3} + 54q^{5} + 88q^{7} - 99q^{9} - 540q^{11} - 418q^{13} + 648q^{15} + 594q^{17} - 836q^{19} + 1056q^{21} + 4104q^{23} + O(q^{25})$

Coefficient field

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

$x - 66$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 16$ )

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	12	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 16, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.107 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, 4, 8, 16]: [1, 2, 4, 7, 12] (certified at ell = 2147483629)
T = [(3, 1), (5, 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(16, 6)[1]   # 16.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 16:6 1000