Atlas › Modular forms › Weight 6 › Level 22 › 22.6.a › 22.6.a.b

Newform orbit 22.6.a.b

$q$ -expansion

$q - 4q^{2} + q^{3} + 16q^{4} - 51q^{5} - 4q^{6} - 166q^{7} - 64q^{8} - 242q^{9} + 204q^{10} - 121q^{11} + 16q^{12} + 692q^{13} + 664q^{14} - 51q^{15} + 256q^{16} - 738q^{17} + 968q^{18} + 1424q^{19} - 816q^{20} - 166q^{21} + 484q^{22} - 1779q^{23} - 64q^{24} + O(q^{25})$

Coefficient field

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

$x + 50$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
11	$+1$

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	1	−51	−166	−121	692	−738	1424	−1779	−2064	6245	−14785	5304	17798	−17184	−30726	−34989	−45940	25343	13311	−53260	77234	55014	125415	−88807

$\operatorname{tr} a_n$ 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 22, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 11, 22]: [1, 2, 6, 17] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (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(22, 6)[1]   # 22.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 22:6 1000