Atlas › Modular forms › Weight 2 › Level 517 › 517.2.a › 517.2.a.f

Newform orbit 517.2.a.f

Trace form

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

$2q + 2q^{2} + 2q^{4} + 4q^{6} - 6q^{7} + 6q^{8} - 2q^{9} + 4q^{10} - 2q^{11} + 8q^{12} - 2q^{13} - 6q^{14} + 4q^{15} + 6q^{16} + 8q^{17} - 2q^{18} + 8q^{19} + 8q^{20} - 2q^{22} - 2q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} - 56x + 712$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
47	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	2	0	0	−6	−2	−2	8	8	−2	−12	6	2	6	16	2	2	16	0	−6	12	4	6	2	−18	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.271 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, 11, 47, 517]: [0, 2, 5, 50] (certified at ell = 2147480791)
T = [(2, 1), (3, 2), (5, 3), (7, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 10 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(517, 2)[5]   # 517.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 517:2 1000