Atlas › Modular forms › Weight 4 › Level 177 › 177.4.a › 177.4.a.d

Newform orbit 177.4.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + 51q^{8} + 72q^{9} + 21q^{10} + 67q^{11} + 102q^{12} + 33q^{13} + 79q^{14} + 126q^{15} - 30q^{16} + 139q^{17} + 54q^{18} + 64q^{19} + 117q^{20} + 159q^{21} - 84q^{22} + 226q^{23} + 153q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} - 48x^{7} + 534x^{6} + 5617x^{5} - 115037x^{4} - 18626x^{3} + 6231436x^{2} - 8667983x - 84466198$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
59	$-1$

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
$\operatorname{tr} a_p$	6	24	42	53	67	33	139	64	226	456	124	127	425	−115	420	98	472	−1254	−1010	−17	−1180	−873	759	988	−668

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.127 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, 3, 59, 177]: [1, 2, 16, 62] (certified at ell = 2147482481)
T = [(2, 1), (5, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 5 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(177, 4)[3]   # 177.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 177:4 1000