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

Newform orbit 57.8.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + q^{2} - 135q^{3} + 553q^{4} + 138q^{5} - 27q^{6} + 670q^{7} - 543q^{8} + 3645q^{9} - 5348q^{10} + 6298q^{11} - 14931q^{12} + 2528q^{13} + 10060q^{14} - 3726q^{15} - 2783q^{16} + 25688q^{17} + 729q^{18} - 34295q^{19} + 193884q^{20} - 18090q^{21} + 248992q^{22} + 210002q^{23} + 14661q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 139x^{4} - 231425x^{3} + 12583319x^{2} + 14249164912x + 450743069492$

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 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
$\operatorname{tr} a_p$	1	−135	138	670	6298	2528	25688	−34295	210002	70210	224162	−751756	385590	172822	654096	−429110	−2127332	−5825040	−1622744	1563936	453092	9589426	33248496	19222206	10677126

$\operatorname{tr} a_n$ 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)[1]   # 57.8.a.b

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

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