Atlas › Modular forms › Weight 8 › Level 5 › 5.8.a › 5.8.a.a

Newform orbit 5.8.a.a

$q$ -expansion

$q - 14q^{2} - 48q^{3} + 68q^{4} + 125q^{5} + 672q^{6} - 1644q^{7} + 840q^{8} + 117q^{9} - 1750q^{10} + 172q^{11} - 3264q^{12} + 3862q^{13} + 23016q^{14} - 6000q^{15} - 20464q^{16} - 12254q^{17} - 1638q^{18} - 25940q^{19} + 8500q^{20} + 78912q^{21} - 2408q^{22} + 12972q^{23} - 40320q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 62$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$

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 odd 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
$\operatorname{tr} a_p$	−14	−48	125	−1644	172	3862	−12254	−25940	12972	−81610	−156888	110126	467882	−499208	−396884	−1280498	−1337420	−923978	−797304	5103392	−4267478	−960	6140832	2010570	−4881934

$\operatorname{tr} a_n$ 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 5, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 5]: [1, 5] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (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(5, 8)[0]   # 5.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 5:8 1000