Atlas › Modular forms › Weight 4 › Level 162 › 162.4.a › 162.4.a.d

Newform orbit 162.4.a.d

$q$ -expansion

$q + 2q^{2} + 4q^{4} - 9q^{5} - 31q^{7} + 8q^{8} - 18q^{10} - 15q^{11} - 37q^{13} - 62q^{14} + 16q^{16} - 42q^{17} - 28q^{19} - 36q^{20} - 30q^{22} + 195q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x + 102$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	— ( $3^2 \mid 162$ )

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
$\operatorname{tr} a_p$	2	0	−9	−31	−15	−37	−42	−28	195	111	−205	−166	−261	−43	177	114	159	191	−421	156	182	1133	−1083	−1050	−901

$\operatorname{tr} a_n$ 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 162, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.153 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, 6, 9, 18, 27, 54, 81, 162]: [1, 2, 2, 5, 4, 11, 10, 29, 28, 83] (certified at ell = 2147483179)
T = [(5, 1), (7, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(162, 4)[3]   # 162.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 162:4 1000