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

Newform orbit 177.4.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 8q^{2} + 21q^{3} + 22q^{4} - 28q^{5} - 24q^{6} - 59q^{7} - 117q^{8} + 63q^{9} - 79q^{10} - 131q^{11} + 66q^{12} - 123q^{13} - 117q^{14} - 84q^{15} + 202q^{16} - 235q^{17} - 72q^{18} - 80q^{19} + 61q^{20} - 177q^{21} + 688q^{22} - 274q^{23} - 351q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} + 36x^{6} + 154x^{5} - 5947x^{4} - 37467x^{3} + 371126x^{2} + 1504000x - 8997599$

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 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	21	−28	−59	−131	−123	−235	−80	−274	−406	−346	−157	−825	−815	−1196	−900	−413	420	1316	−173	−418	2635	457	592	−1906

$\operatorname{tr} a_n$ 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)[0]   # 177.4.a.a

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

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