Atlas › Modular forms › Weight 8 › Level 62 › 62.8.a › 62.8.a.c

Newform orbit 62.8.a.c

Trace form

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

$5q - 40q^{2} + 28q^{3} + 320q^{4} + 152q^{5} - 224q^{6} - 132q^{7} - 2560q^{8} + 9149q^{9} - 1216q^{10} - 4218q^{11} + 1792q^{12} - 7014q^{13} + 1056q^{14} - 30666q^{15} + 20480q^{16} - 30180q^{17} - 73192q^{18} - 8320q^{19} + 9728q^{20} + 117298q^{21} + 33744q^{22} + 67130q^{23} - 14336q^{24} + O(q^{25})$

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

$x^{5} - 180x^{4} - 345943x^{3} + 74916772x^{2} + 29646125654x - 7277639530380$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
31	$+1$

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	28	152	−132	−4218	−7014	−30180	−8320	67130	290034	−148955	682400	197844	2333288	−405160	1029596	−400004	−4933754	2699572	−613944	7894998	10463774	−12217942	22649884	−7610048

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 31, 62]: [1, 3, 19, 58] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 6 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(62, 8)[2]   # 62.8.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 62:8 1000