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

Newform orbit 178.4.a.c

Trace form

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

$6q - 12q^{2} - 9q^{3} + 24q^{4} - 3q^{5} + 18q^{6} - 26q^{7} - 48q^{8} + 59q^{9} + 6q^{10} + 6q^{11} - 36q^{12} - 10q^{13} + 52q^{14} - 81q^{15} + 96q^{16} - 131q^{17} - 118q^{18} + 5q^{19} - 12q^{20} - 12q^{22} - 355q^{23} + 72q^{24} + O(q^{25})$

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

$x^{6} + 12x^{5} - 254x^{4} - 2582x^{3} + 12648x^{2} + 110862x - 74628$

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 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
$\operatorname{tr} a_p$	−12	−9	−3	−26	6	−10	−131	5	−355	−452	−475	−516	−260	−415	−884	−781	394	926	780	−1762	1281	1174	−238	534	−49

$\operatorname{tr} a_n$ 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)[2]   # 178.4.a.c

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

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