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

Newform orbit 22.6.a.c

$q$ -expansion

$q + 4q^{2} - 29q^{3} + 16q^{4} - 31q^{5} - 116q^{6} - 230q^{7} + 64q^{8} + 598q^{9} - 124q^{10} + 121q^{11} - 464q^{12} + 112q^{13} - 920q^{14} + 899q^{15} + 256q^{16} - 1142q^{17} + 2392q^{18} - 612q^{19} - 496q^{20} + 6670q^{21} + 484q^{22} - 1941q^{23} - 1856q^{24} + O(q^{25})$

Coefficient field

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

$x + 60$

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	−29	−31	−230	121	112	−1142	−612	−1941	1192	−1037	8083	−10444	58	8656	−20318	−21351	47044	48093	−24967	−42288	−72410	−15806	−114761	−5159

$\operatorname{tr} a_n$ 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)[2]   # 22.6.a.c

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

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