Atlas › Modular forms › Weight 4 › Level 54 › 54.4.a › 54.4.a.a

Newform orbit 54.4.a.a

$q$ -expansion

$q - 2q^{2} + 4q^{4} - 12q^{5} - 7q^{7} - 8q^{8} + 24q^{10} - 60q^{11} - 79q^{13} + 14q^{14} + 16q^{16} + 108q^{17} + 11q^{19} - 48q^{20} + 120q^{22} + 132q^{23} + O(q^{25})$

Coefficient field

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

$x + 19$

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
$\operatorname{tr} a_p$	−2	0	−12	−7	−60	−79	108	11	132	−96	20	−169	−192	488	−204	−360	−156	83	47	−216	−511	−529	1128	−36	605

$\operatorname{tr} a_n$ 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 54, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 2, 2, 5, 4, 11, 10, 29] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 4)[0]   # 54.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 54:4 1000