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

Newform orbit 158.4.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 6q^{2} - 3q^{3} + 12q^{4} + 6q^{5} + 6q^{6} - 39q^{7} - 24q^{8} + 36q^{9} - 12q^{10} - 5q^{11} - 12q^{12} - 46q^{13} + 78q^{14} + 9q^{15} + 48q^{16} - 120q^{17} - 72q^{18} + 129q^{19} + 24q^{20} - 273q^{21} + 10q^{22} - 69q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 3x^{2} - 73x + 278$

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
$\operatorname{tr} a_p$	−6	−3	6	−39	−5	−46	−120	129	−69	−198	−125	−478	40	−270	629	354	285	−924	−1431	913	−1457	237	1866	334	−3590

$\operatorname{tr} a_n$ 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)[1]   # 158.4.a.b

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

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