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

Newform orbit 367.2.a.a

Trace form

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

$11q - 8q^{2} - 6q^{3} + 10q^{4} - 8q^{5} + q^{6} - 7q^{7} - 18q^{8} - 3q^{9} - q^{10} - 10q^{11} - 10q^{12} - 5q^{13} + q^{14} + q^{15} + 16q^{16} - 32q^{17} - 8q^{18} + 3q^{19} - 13q^{20} - 10q^{21} - 15q^{23} + 19q^{24} + O(q^{25})$

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

$x^{11} + 14x^{10} + 66x^{9} + 93x^{8} - 156x^{7} - 528x^{6} - 181x^{5} + 579x^{4} + 355x^{3} - 247x^{2} - 118x + 55$

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 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
$\operatorname{tr} a_p$	−8	−6	−8	−7	−10	−5	−32	3	−15	−26	3	−10	−32	−12	−9	−32	14	15	23	−4	−10	4	19	−3	−24

$\operatorname{tr} a_n$ 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)[0]   # 367.2.a.a

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

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