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

Newform orbit 24.12.a.c

$q$ -expansion

$q + 243q^{3} + 1870q^{5} - 72312q^{7} + 59049q^{9} + 147940q^{11} - 1562858q^{13} + 454410q^{15} - 145774q^{17} + 1096796q^{19} - 17571816q^{21} - 60014264q^{23} + O(q^{25})$

Coefficient field

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

$x + 215066$

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	1870	−72312	147940	−1562858	−145774	1096796	−60014264	−19626954	−239950480	488238078	47066010	428866948	450903216	4336685950	−8937556460	4673884486	7498937612	−27032101480	11676141658	2478876544	42745596956	−93270772662	118032786914

$\operatorname{tr} a_n$ 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)[2]   # 24.12.a.c

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

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