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

Newform orbit 208.4.a.k

Trace form

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

$2q + 3q^{3} + 23q^{5} - 27q^{7} + 59q^{9} - 2q^{11} + 26q^{13} + 143q^{15} - q^{17} + 42q^{19} - 149q^{21} - 24q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{217})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{2} - 433x - 211412$

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	23	−27	−2	26	−1	42	−24	−60	404	75	506	381	−455	494	−262	−258	638	−1393	−116	104	1392	356	312

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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)[10]   # 208.4.a.k

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

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