Atlas › Modular forms › Weight 8 › Level 58 › 58.8.a › 58.8.a.d

Newform orbit 58.8.a.d

Trace form

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

$5q + 40q^{2} + 56q^{3} + 320q^{4} + 570q^{5} + 448q^{6} + 920q^{7} + 2560q^{8} + 3605q^{9} + 4560q^{10} + 8000q^{11} + 3584q^{12} + 17954q^{13} + 7360q^{14} - 5812q^{15} + 20480q^{16} + 50522q^{17} + 28840q^{18} + 51364q^{19} + 36480q^{20} + 76064q^{21} + 64000q^{22} + 92656q^{23} + 28672q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{5} - 626x^{4} - 74276x^{3} + 97828344x^{2} - 9268454508x - 1401159789576$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
29	$-1$

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
$\operatorname{tr} a_p$	40	56	570	920	8000	17954	50522	51364	92656	121945	77300	94942	−87174	−42128	−986628	662530	−4064436	−2271010	−4281364	−6427216	−10309302	−6405148	5064660	3155210	21643970

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 2, 29, 58]: [1, 3, 19, 55] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(58, 8)[3]   # 58.8.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 58:8 1000