Atlas › Modular forms › Weight 10 › Level 9 › 9.10.a › 9.10.a.c

Newform orbit 9.10.a.c

$q$ -expansion

$q + 36q^{2} + 784q^{4} + 1314q^{5} - 4480q^{7} + 9792q^{8} + 47304q^{10} - 1476q^{11} - 151522q^{13} - 161280q^{14} - 48896q^{16} - 108162q^{17} + 593084q^{19} + 1030176q^{20} - 53136q^{22} + 969480q^{23} + O(q^{25})$

Coefficient field

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

$x - 1350$

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
$\operatorname{tr} a_p$	36	0	1314	−4480	−1476	−151522	−108162	593084	969480	6642522	7070600	−7472410	4350150	−4358716	−28309248	−16111710	86075964	32213918	99531452	44170488	−23560630	−401754760	744528708	−769871034	907130882

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.29 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]: [1, 4, 10] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 3 = 1 x 3 (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, 10)[2]   # 9.10.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 9:10 1000