Atlas › Modular forms › Weight 4 › Level 112 › 112.4.a › 112.4.a.f

Newform orbit 112.4.a.f

$q$ -expansion

$q + 2q^{3} + 16q^{5} + 7q^{7} - 23q^{9} + 8q^{11} + 28q^{13} + 32q^{15} + 54q^{17} + 110q^{19} + 14q^{21} - 48q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ is a root of

$x - 772$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 112$ )
7	$-1$

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	2	16	7	8	28	54	110	−48	−110	−12	−246	182	−128	−324	−162	−810	−488	−244	768	−702	−440	1302	730	294

$\operatorname{tr} a_n$ 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 112, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 4, 7, 8, 14, 16, 28, 56, 112]: [1, 2, 3, 3, 5, 8, 8, 15, 28, 52] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 9 = 1 x 9 (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(112, 4)[5]   # 112.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 112:4 1000