Atlas › Modular forms › Weight 4 › Level 38 › 38.4.a › 38.4.a.b

Newform orbit 38.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} + q^{3} + 8q^{4} + 10q^{5} - 2q^{6} + 57q^{7} - 16q^{8} + 35q^{9} - 20q^{10} + 10q^{11} + 4q^{12} + 13q^{13} - 114q^{14} - 172q^{15} + 32q^{16} - 51q^{17} - 70q^{18} - 38q^{19} + 40q^{20} + 117q^{21} - 20q^{22} - 155q^{23} - 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{177})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 11x - 14$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
19	$+1$

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
$\operatorname{tr} a_p$	−4	1	10	57	10	13	−51	−38	−155	−79	−16	380	−790	296	−200	397	201	−680	−939	406	123	106	2226	−870	−1864

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.1 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 engine's checks

dim M^+ at levels [1, 2, 19, 38]: [1, 2, 6, 17] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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(38, 4)[1]   # 38.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 38:4 1000