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

Newform orbit 19.6.a.b

$q$ -expansion

$q - 2q^{2} - q^{3} - 28q^{4} - 24q^{5} + 2q^{6} - 167q^{7} + 120q^{8} - 242q^{9} + 48q^{10} + 262q^{11} + 28q^{12} + 749q^{13} + 334q^{14} + 24q^{15} + 656q^{16} - 1597q^{17} + 484q^{18} - 361q^{19} + 672q^{20} + 167q^{21} - 524q^{22} - 2011q^{23} - 120q^{24} + O(q^{25})$

Coefficient field

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

$x + 3$

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 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
$\operatorname{tr} a_p$	−2	−1	−24	−167	262	749	−1597	−361	−2011	−1055	−1548	9378	−10248	10544	−6912	−35291	33655	−26218	45083	30942	46969	−64430	−13986	−137700	−22162

$\operatorname{tr} a_n$ 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)[1]   # 19.6.a.b

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

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