Atlas › Modular forms › Weight 6 › Level 62 › 62.6.a › 62.6.a.d

Newform orbit 62.6.a.d

Trace form

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

$4q + 16q^{2} + 28q^{3} + 64q^{4} - 10q^{5} + 112q^{6} + 266q^{7} + 256q^{8} + 908q^{9} - 40q^{10} + 932q^{11} + 448q^{12} + 88q^{13} + 1064q^{14} - 348q^{15} + 1024q^{16} + 392q^{17} + 3632q^{18} - 206q^{19} - 160q^{20} + 2428q^{21} + 3728q^{22} + 1116q^{23} + 1792q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} - 18x^{3} - 7387x^{2} + 83308x + 12334280$

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
$\operatorname{tr} a_p$	16	28	−10	266	932	88	392	−206	1116	−1916	−3844	12104	−35702	−7656	−10088	18572	−7698	−29664	84520	106950	35584	117808	48012	44020	68806

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 2, 15, 42] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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, 6)[3]   # 62.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 62:6 1000