Atlas › Modular forms › Weight 2 › Level 317 › 317.2.a › 317.2.a.b

Newform orbit 317.2.a.b

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + q^{2} + 11q^{3} + 15q^{4} + 2q^{5} + 20q^{7} - 3q^{8} + 16q^{9} - q^{10} + 6q^{11} + 9q^{12} + 14q^{13} - 3q^{14} - 6q^{15} + 7q^{16} + 5q^{17} + q^{18} + 31q^{19} - 6q^{20} + 2q^{21} - 5q^{22} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 12x^{14} + 19x^{13} + 283x^{12} - 1124x^{11} - 1420x^{10} + 12920x^{9} - 8624x^{8} - 50379x^{7} + 77325x^{6} + 45318x^{5} - 129347x^{4} + 13755x^{3} + 58453x^{2} - 8624x - 7517$

Atkin–Lehner signs

$p$	$w_p$
317	$-1$

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
$\operatorname{tr} a_p$	1	11	2	20	6	14	5	31	0	−2	12	14	−4	41	3	−18	−12	10	21	−10	35	−1	25	−27	32

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.175 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, 317]: [0, 27] (certified at ell = 2147479753)
T = [(2, 1), (3, 1)]; h of degree 26 = 1 x 26 (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(317, 2)[1]   # 317.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 317:2 1000