Atlas › Modular forms › Weight 4 › Level 198 › 198.4.a › 198.4.a.a

Newform orbit 198.4.a.a

$q$ -expansion

$q - 2q^{2} + 4q^{4} - 10q^{5} + 16q^{7} - 8q^{8} + 20q^{10} - 11q^{11} + 10q^{13} - 32q^{14} + 16q^{16} + 10q^{17} - 144q^{19} - 40q^{20} + 22q^{22} + 84q^{23} + O(q^{25})$

Coefficient field

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

$x - 158$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	— ( $3^2 \mid 198$ )
11	$+1$

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
$\operatorname{tr} a_p$	−2	0	−10	16	−11	10	10	−144	84	−218	−176	46	26	−488	−404	−194	−444	202	−84	764	354	1312	1252	1222	−1358

$\operatorname{tr} a_n$ 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 198, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.288 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, 6, 9, 11, 18, 22, 33, 66, 99, 198]: [1, 2, 2, 5, 4, 4, 11, 11, 14, 40, 38, 112] (certified at ell = 2147482951)
T = [(5, 1), (7, 3), (13, 5), (17, 7)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 5 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(198, 4)[0]   # 198.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 198:4 1000