Atlas › Modular forms › Weight 4 › Level 22 › 22.4.a › 22.4.a.a

Newform orbit 22.4.a.a

$q$ -expansion

$q - 2q^{2} - 7q^{3} + 4q^{4} - 19q^{5} + 14q^{6} + 14q^{7} - 8q^{8} + 22q^{9} + 38q^{10} + 11q^{11} - 28q^{12} - 72q^{13} - 28q^{14} + 133q^{15} + 16q^{16} - 46q^{17} - 44q^{18} - 20q^{19} - 76q^{20} - 98q^{21} - 22q^{22} - 107q^{23} + 56q^{24} + O(q^{25})$

Coefficient field

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

$x + 26$

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
$\operatorname{tr} a_p$	−2	−7	−19	14	11	−72	−46	−20	−107	120	117	−201	−228	−242	−96	458	435	−668	439	−1113	−72	−70	358	895	409

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ 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 0.099 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, 4, 11] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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, 4)[0]   # 22.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 22:4 1000