Atlas › Modular forms › Weight 2 › Level 395 › 395.2.a › 395.2.a.g

Newform orbit 395.2.a.g

Trace form

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

$4q + q^{2} + 2q^{3} + 7q^{4} + 4q^{5} - 9q^{6} + 3q^{7} + 10q^{9} + q^{10} - 6q^{11} + 11q^{12} + 8q^{13} - 14q^{14} + 2q^{15} + 33q^{16} + 12q^{17} - 15q^{18} - 5q^{19} + 7q^{20} + 11q^{21} - 11q^{22} - 16q^{23} - 19q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} - 3x^{3} - 5x^{2} + 16x - 1$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
79	$+1$

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	2	4	3	−6	8	12	−5	−16	1	−5	−2	−16	4	6	21	19	−7	25	−13	−7	−4	−29	−2	25

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.194 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, 5, 79, 395]: [0, 1, 7, 42] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (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(395, 2)[6]   # 395.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 395:2 1000