Atlas › Modular forms › Weight 8 › Level 6 › 6.8.a › 6.8.a.a

Newform orbit 6.8.a.a

$q$ -expansion

$q + 8q^{2} + 27q^{3} + 64q^{4} - 114q^{5} + 216q^{6} - 1576q^{7} + 512q^{8} + 729q^{9} - 912q^{10} + 7332q^{11} + 1728q^{12} - 3802q^{13} - 12608q^{14} - 3078q^{15} + 4096q^{16} - 6606q^{17} + 5832q^{18} + 24860q^{19} - 7296q^{20} - 42552q^{21} + 58656q^{22} + 41448q^{23} + 13824q^{24} + O(q^{25})$

Coefficient field

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

$x + 1690$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$

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	27	−114	−1576	7332	−3802	−6606	24860	41448	−41610	33152	−36466	−639078	−156412	−433776	786078	745140	−1660618	−3290836	5716152	2659898	3807440	2229468	5991210	−4060126

$\operatorname{tr} a_n$ 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 6, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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]: [1, 3, 3, 9] (certified at ell = 2147483647)
T = [(5, 1), (7, 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(6, 8)[0]   # 6.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 6:8 1000