Atlas › Modular forms › Weight 2 › Level 393 › 393.2.a › 393.2.a.e

Newform orbit 393.2.a.e

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} - 9x^{5} + 22x^{4} - 6x^{3} - 30x^{2} + 17x + 7$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
131	$+1$

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
$\operatorname{tr} a_p$	1	6	8	4	6	−3	4	−1	14	12	−2	−12	7	−5	19	4	5	−20	4	11	−18	−6	13	−8	−14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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, 131, 393]: [0, 1, 12, 46] (certified at ell = 2147482481)
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(393, 2)[4]   # 393.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 393:2 1000