Atlas › Modular forms › Weight 4 › Level 200 › 200.4.a › 200.4.a.b

Newform orbit 200.4.a.b

$q$ -expansion

$q - 9q^{3} - 26q^{7} + 54q^{9} - 59q^{11} - 28q^{13} - 5q^{17} + 109q^{19} + 234q^{21} + 194q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ is a root of

$x - 1381$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 200$ )
5	— ( $5^2 \mid 200$ )

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	−9	0	−26	−59	−28	−5	109	194	−32	10	198	117	−388	68	18	392	−710	253	−612	549	414	121	−81	1502

$\operatorname{tr} a_n$ 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 200, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.45 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, 5, 8, 10, 20, 25, 40, 50, 100, 200]: [1, 2, 3, 3, 5, 7, 12, 9, 22, 25, 48, 94] (certified at ell = 2147482921)
T = [(3, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 6 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(200, 4)[1]   # 200.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 200:4 1000