Atlas › Modular forms › Weight 4 › Level 160 › 160.4.a › 160.4.a.b

Newform orbit 160.4.a.b

$q$ -expansion

$q + 2q^{3} - 5q^{5} - 6q^{7} - 23q^{9} - 60q^{11} + 50q^{13} - 10q^{15} - 30q^{17} - 40q^{19} - 12q^{21} - 178q^{23} + O(q^{25})$

Coefficient field

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

$x + 4$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 160$ )
5	$+1$

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	−5	−6	−60	50	−30	−40	−178	166	−20	10	−250	−142	−214	490	800	250	774	−100	−230	1320	−982	874	−310

$\operatorname{tr} a_n$ 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 160, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 5, 8, 10, 16, 20, 32, 40, 80, 160]: [1, 2, 3, 3, 5, 7, 8, 12, 14, 22, 40, 76] (certified at ell = 2147483497)
T = [(3, 1), (7, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(160, 4)[1]   # 160.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 160:4 1000