Atlas › Modular forms › Weight 4 › Level 208 › 208.4.a › 208.4.a.f

Newform orbit 208.4.a.f

$q$ -expansion

$q + 3q^{3} - 13q^{5} + 11q^{7} - 18q^{9} + 2q^{11} - 13q^{13} - 39q^{15} - 51q^{17} - 150q^{19} + 33q^{21} + 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17} + 9T_{19}$ is a root of

$x + 1733$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 208$ )
13	$+1$

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
$\operatorname{tr} a_p$	0	3	−13	11	2	−13	−51	−150	4	−118	116	63	−288	293	335	−708	−566	904	−382	−7	518	100	1440	1254	1262

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 208, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 13, 16, 26, 52, 104, 208]: [1, 2, 3, 5, 5, 8, 13, 24, 46, 88] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (17, 7), (19, 9)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 7 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(208, 4)[5]   # 208.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 208:4 1000