Atlas › Modular forms › Weight 2 › Level 519 › 519.2.a › 519.2.a.c

Newform orbit 519.2.a.c

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{11} - 12x^{10} + 23x^{9} + 244x^{8} - 1324x^{7} + 1892x^{6} + 1700x^{5} - 6499x^{4} + 2993x^{3} + 4328x^{2} - 3936x + 603$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
173	$+1$

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	11	9	−3	13	−1	18	1	15	6	1	−13	17	−22	14	27	11	6	−21	9	1	−23	5	50	−1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 173, 519]: [0, 1, 15, 60] (certified at ell = 2147482093)
T = [(2, 1), (5, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 3 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(519, 2)[2]   # 519.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 519:2 1000