Atlas › Modular forms › Weight 2 › Level 367 › 367.2.a › 367.2.a.b

Newform orbit 367.2.a.b

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 9q^{2} + 4q^{3} + 21q^{4} + 6q^{5} - q^{6} + q^{7} + 27q^{8} + 29q^{9} - q^{10} + 4q^{11} + 4q^{12} + 5q^{13} + 9q^{14} - 3q^{15} + 21q^{16} + 40q^{17} + 13q^{18} - 3q^{19} - 5q^{20} - 10q^{21} - 2q^{22} + 21q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 13x^{18} + 13x^{17} + 478x^{16} - 1726x^{15} - 5496x^{14} + 34461x^{13} + 7964x^{12} - 287077x^{11} + 258711x^{10} + 1059939x^{9} - 1728454x^{8} - 1288541x^{7} + 3626413x^{6} - 590259x^{5} - 1752368x^{4} + 426879x^{3} + 210706x^{2} - 55183x + 1159$

Atkin–Lehner signs

$p$	$w_p$
367	$-1$

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
$\operatorname{tr} a_p$	9	4	6	1	4	5	40	−3	21	28	−13	8	34	−14	3	34	2	−7	−5	4	12	−28	7	3	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.18 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, 367]: [0, 31] (certified at ell = 2147479381)
T = [(2, 1), (3, 1)]; h of degree 30 = 1 x 30 (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(367, 2)[1]   # 367.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 367:2 1000