Atlas › Modular forms › Weight 8 › Level 48 › 48.8.a › 48.8.a.e

Newform orbit 48.8.a.e

$q$ -expansion

$q + 27q^{3} - 378q^{5} + 832q^{7} + 729q^{9} + 2484q^{11} + 14870q^{13} - 10206q^{15} - 22302q^{17} + 16300q^{19} + 22464q^{21} + 115128q^{23} + O(q^{25})$

Coefficient field

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

$x - 2118$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 48$ )
3	$-1$

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
$\operatorname{tr} a_p$	0	27	−378	832	2484	14870	−22302	16300	115128	157086	16456	−149266	−241110	443188	−922752	−697626	−870156	2067062	1680748	1070280	−2403334	−2301512	−4708692	4143690	−1622974

$\operatorname{tr} a_n$ 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 48, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.328 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, 4, 6, 8, 12, 16, 24, 48]: [1, 3, 3, 5, 9, 9, 17, 16, 32, 60] (certified at ell = 2147483629)
T = [(5, 1), (7, 3)]; h of degree 7 = 1 x 7 (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(48, 8)[4]   # 48.8.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 48:8 1000