Atlas › Modular forms › Weight 4 › Level 178 › 178.4.a › 178.4.a.b

Newform orbit 178.4.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 10q^{2} + 3q^{3} + 20q^{4} + 17q^{5} - 6q^{6} + 2q^{7} - 40q^{8} + 68q^{9} - 34q^{10} + 50q^{11} + 12q^{12} - 10q^{13} - 4q^{14} + 69q^{15} + 80q^{16} + 5q^{17} - 136q^{18} - 71q^{19} + 68q^{20} - 84q^{21} - 100q^{22} + 329q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 20x^{4} - 448x^{3} + 7970x^{2} + 11600x - 12000$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
89	$+1$

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
$\operatorname{tr} a_p$	−10	3	17	2	50	−10	5	−71	329	260	407	286	576	131	1588	1135	−382	−762	−108	−198	−1211	−450	58	−445	−1829

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.126 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, 89, 178]: [1, 2, 24, 70] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 22 = 1 x 22 (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(178, 4)[1]   # 178.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 178:4 1000