Atlas › Modular forms › Weight 4 › Level 247 › 247.4.a › 247.4.a.c

Newform orbit 247.4.a.c

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 9q^{2} - 10q^{3} + 65q^{4} - 34q^{5} - 8q^{6} - 27q^{7} - 138q^{8} + 49q^{9} - 10q^{10} - 57q^{11} - 118q^{12} - 169q^{13} - 96q^{14} - 89q^{15} + 345q^{16} - 329q^{17} - 330q^{18} + 247q^{19} - 453q^{20} + 111q^{21} - 478q^{22} - 380q^{23} - 141q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{13} + 911x^{12} - 284122x^{11} - 411432318x^{10} + 229086630x^{9} + 62997138545708x^{8} + 4560156468926060x^{7} - 4029962463763040747x^{6} - 345017849934175879783x^{5} + 96148434209073470625342x^{4} + 7568869550264821480243628x^{3} - 485135386060116188128953232x^{2} - 24819543134599804990824062320x + 349517582900590961362437518400$

Atkin–Lehner signs

$p$	$w_p$
13	$+1$
19	$-1$

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
$\operatorname{tr} a_p$	−9	−10	−34	−27	−57	−169	−329	247	−380	−73	−266	−418	−1756	−102	−951	−461	−2171	167	753	−1266	−1410	1952	−1521	−1727	−2660

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.254 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 engine's checks

dim M^+ at levels [1, 13, 19, 247]: [1, 5, 6, 72] (certified at ell = 2147483629)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 18 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(247, 4)[2]   # 247.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 247:4 1000