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

Newform orbit 162.4.a.h

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{2} + 8q^{4} + 12q^{5} + 16q^{7} + 16q^{8} + 24q^{10} + 36q^{11} + 10q^{13} + 32q^{14} + 32q^{16} + 120q^{17} - 8q^{19} + 48q^{20} + 72q^{22} + 180q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 60x - 423$

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
$\operatorname{tr} a_p$	4	0	12	16	36	10	120	−8	180	324	−248	−434	192	−608	−384	−408	−1008	742	−104	−1140	850	−440	264	768	−764

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

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)[7]   # 162.4.a.h

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

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