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

Newform orbit 159.4.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 12q^{3} + 6q^{4} + 2q^{5} - 24q^{7} + 6q^{8} + 36q^{9} - 19q^{10} - 20q^{11} - 18q^{12} - 33q^{13} - 113q^{14} - 6q^{15} - 142q^{16} - 103q^{17} - 99q^{19} - 127q^{20} + 72q^{21} - 254q^{22} + 109q^{23} - 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 2x^{3} - 82x^{2} - 267x - 134$

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 odd 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
$\operatorname{tr} a_p$	0	−12	2	−24	−20	−33	−103	−99	109	−401	−175	−399	−308	−44	524	212	1025	−1011	−141	−480	−1354	−911	515	369	−1581

$\operatorname{tr} a_n$ 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)[3]   # 159.4.a.d

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

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