Atlas › Modular forms › Weight 2 › Level 585 › 585.2.a › 585.2.a.l

Newform orbit 585.2.a.l

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ , whose minimal polynomial is

$x^{2} - 88x - 937$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 585$ )
5	$-1$
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
$\operatorname{tr} a_p$	1	0	2	−5	1	−2	−1	−2	9	−6	12	−9	3	−2	14	3	24	−1	2	25	−12	−3	0	9	5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.586 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, 5, 9, 13, 15, 39, 45, 65, 117, 195, 585]: [0, 1, 1, 2, 1, 4, 6, 8, 8, 16, 32, 88] (certified at ell = 2147483629)
T = [(2, 1), (7, 2), (11, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 20 = 1 x 20 (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(585, 2)[11]   # 585.2.a.l

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

cargo run --release -p sagebrush-web --example atlas -- 585:2 1000