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

Newform orbit 393.2.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} + 11x^{3} + 37x^{2} + 29x - 29$

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 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
$\operatorname{tr} a_p$	−3	4	−8	−8	−4	−5	−10	−1	−18	−16	0	4	7	−5	−23	−20	−5	−10	6	9	6	12	5	12	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 0.191 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)[1]   # 393.2.a.b

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

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