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

Newform orbit 381.2.a.d

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} - 3x^{4} - 11x^{3} + 45x^{2} - 35x - 1$

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	−5	1	0	14	−5	4	4	15	9	3	−5	4	10	−4	3	23	−15	18	12	−43	16	11	9	−20

$\operatorname{tr} a_n$ 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)[3]   # 381.2.a.d

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

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