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

Newform orbit 247.4.a.d

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 5q^{2} + 2q^{3} + 69q^{4} + 40q^{5} + 28q^{6} + 75q^{7} + 63q^{8} + 134q^{9} + 50q^{10} + 71q^{11} + 208q^{12} - 182q^{13} - 96q^{14} + 31q^{15} + 361q^{16} + 297q^{17} + 78q^{18} - 266q^{19} + 355q^{20} + 435q^{21} - 258q^{22} + 288q^{23} + 131q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{14} - 1375x^{13} - 43789x^{12} + 756013830x^{11} - 254463228756x^{10} - 79142103564950x^{9} + 51936769711595704x^{8} - 4542886577165929679x^{7} - 1513344642277320403777x^{6} + 242731638146137632427453x^{5} + 11886134258606401384529574x^{4} - 3033421295568098397907121064x^{3} + 6284998450732412875539409248x^{2} + 9076117581499035347773694929296x - 211673689764339133998361577218464$

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 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
$\operatorname{tr} a_p$	5	2	40	75	71	−182	297	−266	288	−59	488	−118	738	168	1491	−143	37	−643	637	700	46	836	325	4441	342

$\operatorname{tr} a_n$ 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)[3]   # 247.4.a.d

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

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