Atlas › Modular forms › Weight 8 › Level 54 › 54.8.a › 54.8.a.h

Newform orbit 54.8.a.h

Trace form

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

$2q + 16q^{2} + 128q^{4} - 48q^{5} + 880q^{7} + 1024q^{8} - 384q^{10} + 7230q^{11} + 8560q^{13} + 7040q^{14} + 8192q^{16} + 25704q^{17} + 37048q^{19} - 3072q^{20} + 57840q^{22} - 59628q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{329})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 2592x + 720252$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	— ( $3^2 \mid 54$ )

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
$\operatorname{tr} a_p$	16	0	−48	880	7230	8560	25704	37048	−59628	−275280	245584	−71060	494520	825280	−927708	−1382112	1588920	−3021968	−1882160	−3394440	895090	−369920	−9352050	−912960	1359790

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 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]: [1, 3, 3, 9, 8, 23, 22, 65] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; 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(54, 8)[7]   # 54.8.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 54:8 1000