Atlas › Modular forms › Weight 2 › Level 621 › 621.2.a › 621.2.a.e

Newform orbit 621.2.a.e

Trace form

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

$2q - q^{2} + 5q^{4} + q^{5} - q^{7} - 9q^{8} - 9q^{10} - 9q^{11} - 6q^{13} - 8q^{14} + 3q^{16} - 9q^{17} - q^{19} + 11q^{20} + 13q^{22} + 2q^{23} + O(q^{25})$

Coefficient field

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

$x^{2} - 2x - 16$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 621$ )
23	$-1$

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
$\operatorname{tr} a_p$	−1	0	1	−1	−9	−6	−9	−1	2	0	−3	5	−4	0	−6	7	−14	13	11	−4	16	−31	−4	9	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.343 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, 3, 9, 23, 27, 69, 207, 621]: [0, 1, 2, 3, 4, 10, 26, 74] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 5 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(621, 2)[4]   # 621.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 621:2 1000