Atlas › Modular forms › Weight 2 › Level 391 › 391.2.a › 391.2.a.e

Newform orbit 391.2.a.e

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 4q^{2} + 2q^{3} + 16q^{4} + 5q^{5} - 3q^{6} - 9q^{7} + 6q^{8} + 30q^{9} - 9q^{10} + 11q^{11} + 11q^{12} + 7q^{13} + 21q^{14} - 4q^{15} + 20q^{16} + 12q^{17} + 19q^{18} + 4q^{19} - 13q^{20} - 8q^{22} - 12q^{23} - 36q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 6x^{11} - 32x^{10} + 206x^{9} + 353x^{8} - 2571x^{7} - 1212x^{6} + 13891x^{5} - 3342x^{4} - 28980x^{3} + 21150x^{2} + 7957x - 7213$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$
23	$+1$

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
$\operatorname{tr} a_p$	4	2	5	−9	11	7	12	4	−12	22	−4	−15	32	−18	−11	32	25	−11	−22	8	−18	−21	−4	42	−43

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 17, 23, 391]: [0, 2, 3, 38] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (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(391, 2)[4]   # 391.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 391:2 1000