Atlas › Modular forms › Weight 8 › Level 57 › 57.8.a › 57.8.a.a

Newform orbit 57.8.a.a

Trace form

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

$4q - 7q^{2} + 108q^{3} + 77q^{4} + 32q^{5} - 189q^{6} - 2074q^{7} - 3921q^{8} + 2916q^{9} - 4096q^{10} - 6918q^{11} + 2079q^{12} - 11222q^{13} + 10764q^{14} + 864q^{15} + 12113q^{16} - 16866q^{17} - 5103q^{18} - 27436q^{19} - 14996q^{20} - 55998q^{21} - 54244q^{22} - 69310q^{23} - 105867q^{24} + O(q^{25})$

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

$x^{4} - 25x^{3} - 113025x^{2} + 3808981x + 224438108$

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
$\operatorname{tr} a_p$	−7	108	32	−2074	−6918	−11222	−16866	−27436	−69310	−149576	−480374	−574418	103520	−746830	−874584	61584	3105264	4324794	−449932	615432	−6086318	−4300150	1764560	−716200	−17999700

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 12, 48] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 7 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, 8)[0]   # 57.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 57:8 1000