Atlas › Modular forms › Weight 8 › Level 50 › 50.8.a › 50.8.a.g

Newform orbit 50.8.a.g

$q$ -expansion

$q + 8q^{2} - 12q^{3} + 64q^{4} - 96q^{6} - 1016q^{7} + 512q^{8} - 2043q^{9} + 1092q^{11} - 768q^{12} - 1382q^{13} - 8128q^{14} + 4096q^{16} - 14706q^{17} - 16344q^{18} - 39940q^{19} + 12192q^{21} + 8736q^{22} - 68712q^{23} - 6144q^{24} + O(q^{25})$

Coefficient field

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

$x + 1028$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	— ( $5^2 \mid 50$ )

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	−12	0	−1016	1092	−1382	−14706	−39940	−68712	−102570	227552	−160526	10842	630748	−472656	1494018	2640660	827702	126004	−1414728	−980282	−3566800	−5672892	−11951190	−8682146

$\operatorname{tr} a_n$ 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 50, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 5, 10, 25, 50]: [1, 3, 5, 13, 19, 55] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 12 = 1 x 12 (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(50, 8)[6]   # 50.8.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 50:8 1000