Atlas › Modular forms › Weight 10 › Level 24 › 24.10.a › 24.10.a.a

Newform orbit 24.10.a.a

$q$ -expansion

$q - 81q^{3} + 830q^{5} + 672q^{7} + 6561q^{9} - 73468q^{11} - 78242q^{13} - 67230q^{15} - 161726q^{17} - 653572q^{19} - 54432q^{21} - 1066696q^{23} + O(q^{25})$

Coefficient field

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

$x - 1502$

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	−81	830	672	−73468	−78242	−161726	−653572	−1066696	3824838	−1579480	16015590	26268282	−44495228	14324160	−24386050	11942084	−189740258	−106709572	302754376	81769546	315315352	752833276	−433284294	1282496642

$\operatorname{tr} a_n$ 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 24, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.325 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]: [1, 3, 4, 6, 11, 11, 21, 40] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; 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, 10)[0]   # 24.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 24:10 1000