Atlas › Modular forms › Weight 8 › Level 2 › 2.8.a › 2.8.a.a

Newform orbit 2.8.a.a

$q$ -expansion

$q - 8q^{2} + 12q^{3} + 64q^{4} - 210q^{5} - 96q^{6} + 1016q^{7} - 512q^{8} - 2043q^{9} + 1680q^{10} + 1092q^{11} + 768q^{12} + 1382q^{13} - 8128q^{14} - 2520q^{15} + 4096q^{16} + 14706q^{17} + 16344q^{18} - 39940q^{19} - 13440q^{20} + 12192q^{21} - 8736q^{22} + 68712q^{23} - 6144q^{24} + O(q^{25})$

Coefficient field

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

$x + 198$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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	−210	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 2, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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]: [1, 3] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 1 = 1 x 1 (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(2, 8)[0]   # 2.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 2:8 1000