Atlas › Modular forms › Weight 4 › Level 159 › 159.4.a › 159.4.a.a

Newform orbit 159.4.a.a

$q$ -expansion

$q - 5q^{2} - 3q^{3} + 17q^{4} - 21q^{5} + 15q^{6} - 16q^{7} - 45q^{8} + 9q^{9} + 105q^{10} - 45q^{11} - 51q^{12} - 66q^{13} + 80q^{14} + 63q^{15} + 89q^{16} - 123q^{17} - 45q^{18} - 62q^{19} - 357q^{20} + 48q^{21} + 225q^{22} + 47q^{23} + 135q^{24} + O(q^{25})$

Coefficient field

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

$x + 26$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
53	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	−5	−3	−21	−16	−45	−66	−123	−62	47	76	−17	−370	324	343	−394	−53	−379	−398	−234	56	−142	305	−434	−565	−581

$\operatorname{tr} a_n$ 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 159, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.124 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, 53, 159]: [1, 2, 15, 56] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 26 = 1 x 26 (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(159, 4)[0]   # 159.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 159:4 1000