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

Newform orbit 62.8.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 48q^{2} + 40q^{3} + 384q^{4} + 166q^{5} + 320q^{6} + 1796q^{7} + 3072q^{8} + 3858q^{9} + 1328q^{10} + 4470q^{11} + 2560q^{12} + 25312q^{13} + 14368q^{14} + 49738q^{15} + 24576q^{16} + 9142q^{17} + 30864q^{18} + 24436q^{19} + 10624q^{20} + 97970q^{21} + 35760q^{22} + 135830q^{23} + 20480q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 206x^{5} - 432723x^{4} + 79925822x^{3} + 38170622654x^{2} - 8313065403912x + 404457085305912$

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
$\operatorname{tr} a_p$	48	40	166	1796	4470	25312	9142	24436	135830	68680	178746	318014	122158	349652	−471976	−1326742	−2158048	−1225092	−2850752	−8495528	−11470736	−3931382	807306	7277150	19137890

$\operatorname{tr} a_n$ 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)[3]   # 62.8.a.d

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

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