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

Newform orbit 208.4.a.b

$q$ -expansion

$q - 4q^{3} - 18q^{5} - 20q^{7} - 11q^{9} + 48q^{11} + 13q^{13} + 72q^{15} + 66q^{17} + 16q^{19} + 80q^{21} - 168q^{23} + O(q^{25})$

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

$x - 788$

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	−4	−18	−20	48	13	66	16	−168	6	−20	254	−390	124	468	558	96	−826	160	420	362	−776	0	1626	−1294

$\operatorname{tr} a_n$ 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)[1]   # 208.4.a.b

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

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