Atlas › Modular forms › Weight 4 › Level 158 › 158.4.a › 158.4.a.d

Newform orbit 158.4.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 12q^{2} + 6q^{3} + 24q^{4} - 3q^{5} - 12q^{6} + 40q^{7} - 48q^{8} + 72q^{9} + 6q^{10} - 57q^{11} + 24q^{12} + 115q^{13} - 80q^{14} + 12q^{15} + 96q^{16} + 100q^{17} - 144q^{18} + 131q^{19} - 12q^{20} + 302q^{21} + 114q^{22} + 61q^{23} - 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 3x^{5} - 691x^{4} + 4522x^{3} + 97596x^{2} - 816480x - 23328$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
79	$+1$

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
$\operatorname{tr} a_p$	−12	6	−3	40	−57	115	100	131	61	224	947	944	100	690	−162	−936	−1994	−360	−839	−512	1257	−474	−736	−1549	2495

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.122 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, 79, 158]: [1, 2, 21, 62] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (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(158, 4)[3]   # 158.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 158:4 1000