Atlas › Modular forms › Weight 8 › Level 3 › 3.8.a › 3.8.a.a

Newform orbit 3.8.a.a

$q$ -expansion

$q + 6q^{2} - 27q^{3} - 92q^{4} + 390q^{5} - 162q^{6} - 64q^{7} - 1320q^{8} + 729q^{9} + 2340q^{10} - 948q^{11} + 2484q^{12} - 5098q^{13} - 384q^{14} - 10530q^{15} + 3856q^{16} + 28386q^{17} + 4374q^{18} - 8620q^{19} - 35880q^{20} + 1728q^{21} - 5688q^{22} - 15288q^{23} + 35640q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x - 396$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	6	−27	390	−64	−948	−5098	28386	−8620	−15288	36510	−276808	268526	−629718	685772	583296	−428058	1306380	300662	−507244	5560632	1369082	−6913720	−4376748	−8528310	−8826814

$\operatorname{tr} a_n$ 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 3, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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, 3]: [1, 3] (certified at ell = 2147483647)
T = [(2, 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(3, 8)[0]   # 3.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 3:8 1000