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

Newform orbit 393.2.a.c

Trace form

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

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

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

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

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
$\operatorname{tr} a_p$	−1	−4	0	−8	2	−9	−6	−7	−8	10	−10	−10	7	−17	−13	0	−9	2	2	−7	−2	−10	−9	22	−22

$\operatorname{tr} a_n$ 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)[2]   # 393.2.a.c

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

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