Atlas › Modular forms › Weight 6 › Level 38 › 38.6.a › 38.6.a.b

Newform orbit 38.6.a.b

$q$ -expansion

$q + 4q^{2} - 14q^{3} + 16q^{4} - 45q^{5} - 56q^{6} - 121q^{7} + 64q^{8} - 47q^{9} - 180q^{10} - 381q^{11} - 224q^{12} - 100q^{13} - 484q^{14} + 630q^{15} + 256q^{16} + 933q^{17} - 188q^{18} + 361q^{19} - 720q^{20} + 1694q^{21} - 1524q^{22} - 552q^{23} - 896q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 59$

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 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
$\operatorname{tr} a_p$	4	−14	−45	−121	−381	−100	933	361	−552	2394	−4024	9182	−2250	−23377	−26595	−16008	−126	21335	−51760	8574	11153	−1660	95964	118848	−153760

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 38, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 10, 27] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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, 6)[1]   # 38.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 38:6 1000