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

Newform orbit 317.2.a.a

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 14x^{10} + 66x^{9} + 93x^{8} - 163x^{7} - 613x^{6} - 533x^{5} + 26x^{4} + 179x^{3} + 14x^{2} - 14x + 1$

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 odd 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	−4	−20	−10	−16	−7	−31	0	12	−12	−16	8	−47	−9	8	4	−8	−51	26	−31	−9	−39	3	−22

$\operatorname{tr} a_n$ 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)[0]   # 317.2.a.a

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

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