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

Newform orbit 517.2.a.k

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 3q^{2} + 2q^{3} + 15q^{4} + 4q^{5} - 10q^{6} + 7q^{7} + 9q^{8} + 10q^{9} + 2q^{10} + 10q^{11} + 13q^{12} - 11q^{13} + 12q^{14} + 8q^{15} + 9q^{16} - 6q^{17} - 4q^{18} - 4q^{19} + 16q^{20} + 14q^{21} + 3q^{22} + 33q^{23} - 23q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. 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

$x^{10} - 192x^{9} + 605x^{8} + 1343688x^{7} - 17838964x^{6} - 3429238455x^{5} + 9281823568x^{4} + 2537190724103x^{3} + 38120889786x^{2} - 454841772377452x + 1264259755702072$

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
$\operatorname{tr} a_p$	3	2	4	7	10	−11	−6	−4	33	8	−21	−5	−15	8	−10	45	32	−10	13	−20	−36	1	31	−3	45

$\operatorname{tr} a_n$ 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)[10]   # 517.2.a.k

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

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