Atlas › Modular forms › Weight 2 › Level 381 › 381.2.a › 381.2.a.e

Newform orbit 381.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 2q^{2} + 9q^{3} + 14q^{4} - 4q^{5} - 2q^{6} + 10q^{7} - 6q^{8} + 9q^{9} - 4q^{10} + 8q^{11} + 14q^{12} + 14q^{13} + 4q^{14} - 4q^{15} + 32q^{16} - 6q^{17} - 2q^{18} + 12q^{19} - 28q^{20} + 10q^{21} - 18q^{22} - 4q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

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

$x^{9} + 6x^{8} - 27x^{7} - 136x^{6} + 287x^{5} + 877x^{4} - 1228x^{3} - 1253x^{2} + 620x + 77$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
127	$+1$

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
$\operatorname{tr} a_p$	−2	9	−4	10	8	14	−6	12	−4	−8	4	22	−2	6	−2	−12	−6	2	18	24	14	12	−20	−30	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 0.19 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, 3, 127, 381]: [0, 1, 11, 44] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 21 = 1 x 21 (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(381, 2)[4]   # 381.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 381:2 1000