Atlas › Modular forms › Weight 12 › Level 24 › 24.12.a › 24.12.a.a

Newform orbit 24.12.a.a

$q$ -expansion

$q - 243q^{3} - 7130q^{5} - 19536q^{7} + 59049q^{9} - 196148q^{11} + 361414q^{13} + 1732590q^{15} - 130942q^{17} + 18516692q^{19} + 4747248q^{21} + 21560872q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x + 65738$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 24$ )
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	−243	−7130	−19536	−196148	361414	−130942	18516692	21560872	191663742	207933800	−200784930	−1435256598	712703116	−496082400	−3350114330	4583222956	3427501702	17079378356	−7915078504	31559658778	41023578808	−19974672172	−10640163606	6441105794

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

Sato–Tate

The normalized $a_p / 2p^{11/2}$ for the primes $p < 1000$ not dividing 24, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.383 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, 8, 12, 24]: [2, 4, 5, 7, 13, 13, 25, 48] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 3 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(24, 12)[0]   # 24.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 24:12 1000