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

Newform orbit 159.4.a.e

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 5q^{2} - 24q^{3} + 29q^{4} + 23q^{5} - 15q^{6} + 20q^{7} + 51q^{8} + 72q^{9} - 44q^{10} + 47q^{11} - 87q^{12} + 137q^{13} + 86q^{14} - 69q^{15} + 385q^{16} + 126q^{17} + 45q^{18} + 319q^{19} + 355q^{20} - 60q^{21} + 609q^{22} + 136q^{23} - 153q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} - 28x^{7} - 267x^{6} + 11155x^{5} - 6153x^{4} - 1082810x^{3} + 2953046x^{2} + 8919135x - 25355655$

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	−24	23	20	47	137	126	319	136	103	408	755	212	−83	−1390	−424	448	2133	−247	−252	2658	2654	−1509	704	2536

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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)[4]   # 159.4.a.e

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

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