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

Newform orbit 158.4.a.c

Trace form

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

$4q + 8q^{2} - 12q^{3} + 16q^{4} - 17q^{5} - 24q^{6} - 58q^{7} + 32q^{8} - 14q^{9} - 34q^{10} - 59q^{11} - 48q^{12} - 99q^{13} - 116q^{14} - 118q^{15} + 64q^{16} - 166q^{17} - 28q^{18} - 219q^{19} - 68q^{20} + 188q^{21} - 118q^{22} - 51q^{23} - 96q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 29x^{3} + 145x^{2} - 1458x - 7996$

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 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
$\operatorname{tr} a_p$	8	−12	−17	−58	−59	−99	−166	−219	−51	−200	129	−70	206	−192	−172	1392	−12	−638	233	−62	−1983	−316	−592	2323	−1089

$\operatorname{tr} a_n$ 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)[2]   # 158.4.a.c

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

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