Atlas › Modular forms › Weight 4 › Level 169 › 169.4.a › 169.4.a.a

Newform orbit 169.4.a.a

$q$ -expansion

$q - 4q^{2} + 2q^{3} + 8q^{4} - 17q^{5} - 8q^{6} - 20q^{7} - 23q^{9} + 68q^{10} + 32q^{11} + 16q^{12} + 80q^{14} - 34q^{15} - 64q^{16} - 13q^{17} + 92q^{18} - 30q^{19} - 136q^{20} - 40q^{21} - 128q^{22} + 78q^{23} + O(q^{25})$

Coefficient field

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

$x - 2$

Atkin–Lehner signs

$p$	$w_p$
13	— ( $13^2 \mid 169$ )

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
$\operatorname{tr} a_p$	−4	2	−17	−20	32	0	−13	−30	78	197	74	227	165	−156	162	93	864	145	−862	−654	−215	−76	−628	266	−238

$\operatorname{tr} a_n$ 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 169, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.125 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, 13, 169]: [1, 5, 47] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 6 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(169, 4)[0]   # 169.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 169:4 1000