Atlas › Modular forms › Weight 4 › Level 168 › 168.4.a › 168.4.a.f

Newform orbit 168.4.a.f

$q$ -expansion

$q + 3q^{3} - 2q^{5} - 7q^{7} + 9q^{9} + 52q^{11} + 86q^{13} - 6q^{15} - 30q^{17} - 4q^{19} - 21q^{21} + 120q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19} + 11T_{23}$ is a root of

$x - 1658$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 168$ )
3	$-1$
7	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	3	−2	−7	52	86	−30	−4	120	246	80	−290	−374	164	464	−162	180	−666	−628	296	−518	−1184	220	−774	−1086

$\operatorname{tr} a_n$ 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 168, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.39 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, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168]: [1, 2, 2, 3, 5, 3, 5, 9, 8, 10, 16, 15, 28, 28, 54, 104] (certified at ell = 2147483647)
T = [(5, 1), (11, 3), (13, 5), (17, 7), (19, 9), (23, 11)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 5 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(168, 4)[5]   # 168.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 168:4 1000