Atlas › Modular forms › Weight 10 › Level 12 › 12.10.a › 12.10.a.a

Newform orbit 12.10.a.a

$q$ -expansion

$q - 81q^{3} + 990q^{5} + 8576q^{7} + 6561q^{9} + 70596q^{11} - 2530q^{13} - 80190q^{15} - 200574q^{17} - 695620q^{19} - 694656q^{21} + 2472696q^{23} + O(q^{25})$

Coefficient field

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

$x - 9566$

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	−81	990	8576	70596	−2530	−200574	−695620	2472696	5474214	3732104	−21898522	−23818950	10612676	2398464	−8994978	−143417916	−19804258	−165625156	−194801400	148729418	−30134152	302054076	909502650	−872463358

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{9/2}$ 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 0.296 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, 4, 6, 11, 21] (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(12, 10)[0]   # 12.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 12:10 1000