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

Newform orbit 22.8.a.b

$q$ -expansion

$q - 8q^{2} + 91q^{3} + 64q^{4} + 185q^{5} - 728q^{6} - 722q^{7} - 512q^{8} + 6094q^{9} - 1480q^{10} - 1331q^{11} + 5824q^{12} + 11020q^{13} + 5776q^{14} + 16835q^{15} + 4096q^{16} - 17210q^{17} - 48752q^{18} - 9288q^{19} + 11840q^{20} - 65702q^{21} + 10648q^{22} + 22971q^{23} - 46592q^{24} + O(q^{25})$

Coefficient field

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

$x - 646$

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 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
$\operatorname{tr} a_p$	−8	91	185	−722	−1331	11020	−17210	−9288	22971	134272	−287765	−316397	−335968	−858110	587680	−244238	−163287	2297260	−3428283	1542953	2216316	1526014	1650370	5760847	−5750759

$\operatorname{tr} a_n$ 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)[1]   # 22.8.a.b

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

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