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

Newform orbit 200.4.a.k

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{3} - 4q^{7} + 2q^{9} + 40q^{11} - 104q^{13} - 96q^{17} + 40q^{19} - 136q^{21} - 284q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{6})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ , whose minimal polynomial is

$x^{2} + 3656x + 2948368$

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	−4	0	−4	40	−104	−96	40	−284	−140	192	−200	−524	−372	−84	296	696	−692	−316	688	656	−736	1628	−660	−896

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

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)[10]   # 200.4.a.k

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

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