Atlas › Modular forms › Weight 6 › Level 54 › 54.6.a › 54.6.a.d

Newform orbit 54.6.a.d

$q$ -expansion

$q + 4q^{2} + 16q^{4} - 84q^{5} - 193q^{7} + 64q^{8} - 336q^{10} - 348q^{11} + 845q^{13} - 772q^{14} + 256q^{16} - 1692q^{17} - 79q^{19} - 1344q^{20} - 1392q^{22} + 564q^{23} + O(q^{25})$

Coefficient field

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

$x + 277$

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
$\operatorname{tr} a_p$	4	0	−84	−193	−348	845	−1692	−79	564	−6432	4940	−3805	12480	−4936	−8124	33192	−42492	−17833	−67699	−28152	−13975	−83983	33384	−77868	−2083

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.133 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, 3, 7, 6, 17, 16, 47] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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, 6)[3]   # 54.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 54:6 1000