Atlas › Modular forms › Weight 4 › Level 171 › 171.4.a › 171.4.a.a

Newform orbit 171.4.a.a

$q$ -expansion

$q - 3q^{2} + q^{4} + 6q^{5} - 16q^{7} + 21q^{8} - 18q^{10} + 18q^{11} - 16q^{13} + 48q^{14} - 71q^{16} + 24q^{17} + 19q^{19} + 6q^{20} - 54q^{22} - 60q^{23} + O(q^{25})$

Coefficient field

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

$x - 3$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 171$ )
19	$-1$

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
$\operatorname{tr} a_p$	−3	0	6	−16	18	−16	24	19	−60	−186	−214	−196	−282	20	−240	210	240	−250	632	168	−538	−142	−126	1470	434

$\operatorname{tr} a_n$ 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 171, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.13 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, 19, 57, 171]: [1, 2, 4, 6, 22, 62] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (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(171, 4)[0]   # 171.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 171:4 1000