Atlas › Modular forms › Weight 8 › Level 22 › 22.8.a › 22.8.a.a

Newform orbit 22.8.a.a

$q$ -expansion

$q - 8q^{2} - 19q^{3} + 64q^{4} + 317q^{5} + 152q^{6} - 1030q^{7} - 512q^{8} - 1826q^{9} - 2536q^{10} + 1331q^{11} - 1216q^{12} - 14676q^{13} + 8240q^{14} - 6023q^{15} + 4096q^{16} - 30058q^{17} + 14608q^{18} + 38056q^{19} + 20288q^{20} + 19570q^{21} - 10648q^{22} - 12911q^{23} + 9728q^{24} + O(q^{25})$

Coefficient field

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

$x - 932$

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
$\operatorname{tr} a_p$	−8	−19	317	−1030	1331	−14676	−30058	38056	−12911	−90480	−139023	251511	−318192	672430	−519096	773570	2194167	3163180	−1293557	−1207245	−4724772	−2638102	−4830962	−2448233	3948601

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ 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 0.265 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, 3, 8, 23] (certified at ell = 2147483171)
T = [(3, 1), (5, 3)]; 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, 8)[0]   # 22.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 22:8 1000