Atlas › Modular forms › Weight 8 › Level 12 › 12.8.a › 12.8.a.a

Newform orbit 12.8.a.a

$q$ -expansion

$q - 27q^{3} - 378q^{5} - 832q^{7} + 729q^{9} - 2484q^{11} + 14870q^{13} + 10206q^{15} - 22302q^{17} - 16300q^{19} + 22464q^{21} - 115128q^{23} + O(q^{25})$

Coefficient field

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

$x + 1210$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 12$ )
3	$+1$

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
$\operatorname{tr} a_p$	0	−27	−378	−832	−2484	14870	−22302	−16300	−115128	157086	−16456	−149266	−241110	−443188	922752	−697626	870156	2067062	−1680748	−1070280	−2403334	2301512	4708692	4143690	−1622974

$\operatorname{tr} a_n$ 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 12, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 4, 6, 12]: [1, 3, 3, 5, 9, 17] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 2 = 1 x 2 (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(12, 8)[0]   # 12.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 12:8 1000