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

Newform orbit 54.8.a.e

$q$ -expansion

$q + 8q^{2} + 64q^{4} - 105q^{5} - 937q^{7} + 512q^{8} - 840q^{10} - 5943q^{11} + 68q^{13} - 7496q^{14} + 4096q^{16} + 5400q^{17} - 48382q^{19} - 6720q^{20} - 47544q^{22} + 642q^{23} + O(q^{25})$

Coefficient field

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

$x + 2916$

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
$\operatorname{tr} a_p$	8	0	−105	−937	−5943	68	5400	−48382	642	125934	−161275	−414286	627474	570590	−538698	−356283	2910828	2684168	2681078	3705480	−153151	−7579288	−9345999	−4033602	−5754097

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ 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 0.342 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)[4]   # 54.8.a.e

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

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