Atlas › Modular forms › Weight 2 › Level 624 › 624.2.a › 624.2.a.k

Newform orbit 624.2.a.k

Trace form

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

$2q - 2q^{3} + 2q^{9} + 4q^{11} - 2q^{13} + 4q^{17} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{17} + 9T_{19} + 11T_{23}$ , whose minimal polynomial is

$x^{2} - 136x + 1736$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 624$ )
3	$+1$
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	−2	0	0	4	−2	4	0	8	4	8	−4	16	−8	12	−4	−4	4	−8	−4	12	0	4	24	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.59 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624]: [0, 1, 1, 2, 3, 3, 5, 1, 4, 8, 5, 6, 12, 10, 18, 18, 34, 32, 64, 120] (certified at ell = 2147483629)
T = [(5, 1), (7, 3), (11, 5), (17, 7), (19, 9), (23, 11)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 4 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(624, 2)[10]   # 624.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 624:2 1000