Atlas › Modular forms › Weight 12 › Level 9 › 9.12.a › 9.12.a.b

Newform orbit 9.12.a.b

$q$ -expansion

$q + 24q^{2} - 1472q^{4} - 4830q^{5} - 16744q^{7} - 84480q^{8} - 115920q^{10} - 534612q^{11} - 577738q^{13} - 401856q^{14} + 987136q^{16} + 6905934q^{17} + 10661420q^{19} + 7109760q^{20} - 12830688q^{22} - 18643272q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x + 4806$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 9$ )

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
$\operatorname{tr} a_p$	24	0	−4830	−16744	−534612	−577738	6905934	10661420	−18643272	−128406630	−52843168	−182213314	−308120442	−17125708	−2687348496	1596055698	5189203740	6956478662	−15481826884	−9791485272	1463791322	38116845680	29335099668	24992917110	75013568546

$\operatorname{tr} a_n$ 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 9, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 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, 3, 9]: [2, 5, 12] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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(9, 12)[1]   # 9.12.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 9:12 1000