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

Newform orbit 38.6.a.a

$q$ -expansion

$q - 4q^{2} - 6q^{3} + 16q^{4} + 31q^{5} + 24q^{6} - 27q^{7} - 64q^{8} - 207q^{9} - 124q^{10} - 323q^{11} - 96q^{12} - 676q^{13} + 108q^{14} - 186q^{15} + 256q^{16} - 1107q^{17} + 828q^{18} - 361q^{19} + 496q^{20} + 162q^{21} + 1292q^{22} + 1384q^{23} + 384q^{24} + O(q^{25})$

Coefficient field

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

$x - 25$

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	−6	31	−27	−323	−676	−1107	−361	1384	2870	1372	−7982	1202	−9911	3463	17764	27270	20867	15228	40642	−66711	68960	−12396	41220	−113432

$\operatorname{tr} a_n$ 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)[0]   # 38.6.a.a

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

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