Atlas › Modular forms › Weight 6 › Level 57 › 57.6.a › 57.6.a.f

Newform orbit 57.6.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + q^{2} - 36q^{3} + 53q^{4} - 8q^{5} - 9q^{6} - 142q^{7} - 147q^{8} + 324q^{9} - 496q^{10} - 714q^{11} - 477q^{12} - 74q^{13} - 2790q^{14} + 72q^{15} - 2839q^{16} - 3690q^{17} + 81q^{18} - 1444q^{19} - 5158q^{20} + 1278q^{21} - 1582q^{22} + 862q^{23} + 1323q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} + 7x^{3} - 7461x^{2} - 75769x + 7767098$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
19	$+1$

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
$\operatorname{tr} a_p$	1	−36	−8	−142	−714	−74	−3690	−1444	862	992	−7382	19114	−5792	−18634	−35424	−20256	−76944	11562	−110044	−71184	−138830	−59470	−191600	158920	47460

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 19, 57]: [1, 3, 10, 36] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (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(57, 6)[5]   # 57.6.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 57:6 1000