Atlas › Modular forms › Weight 4 › Level 117 › 117.4.a › 117.4.a.a

Newform orbit 117.4.a.a

$q$ -expansion

$q - 8q^{4} + 12q^{5} + 2q^{7} + 36q^{11} + 13q^{13} + 64q^{16} + 78q^{17} + 74q^{19} - 96q^{20} + 96q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{5}$ is a root of

$x - 36$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 117$ )
13	$-1$

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	0	12	2	36	13	78	74	96	−18	−214	−286	384	524	−300	−558	−576	74	38	456	−682	704	888	1020	110

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 117, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.118 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, 13, 39, 117]: [1, 2, 4, 5, 16, 44] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(117, 4)[0]   # 117.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 117:4 1000