Atlas › Modular forms › Weight 6 › Level 19 › 19.6.a › 19.6.a.a

Newform orbit 19.6.a.a

$q$ -expansion

$q - 6q^{2} + 4q^{3} + 4q^{4} + 54q^{5} - 24q^{6} + 248q^{7} + 168q^{8} - 227q^{9} - 324q^{10} + 204q^{11} + 16q^{12} - 370q^{13} - 1488q^{14} + 216q^{15} - 1136q^{16} + 1554q^{17} + 1362q^{18} + 361q^{19} + 216q^{20} + 992q^{21} - 1224q^{22} - 408q^{23} + 672q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 2$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	−6	4	54	248	204	−370	1554	361	−408	6174	−7840	−5146	−7830	−12532	2592	−20778	18972	−18418	−11548	−72984	59114	−44752	−27660	20730	14018

$\operatorname{tr} a_n$ 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 19, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.211 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, 19]: [1, 10] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (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(19, 6)[0]   # 19.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 19:6 1000