Atlas › Modular forms › Weight 6 › Level 86 › 86.6.a › 86.6.a.d

Newform orbit 86.6.a.d

Trace form

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

$6q + 24q^{2} + 17q^{3} + 96q^{4} + 61q^{5} + 68q^{6} + 210q^{7} + 384q^{8} + 749q^{9} + 244q^{10} + 721q^{11} + 272q^{12} + 1987q^{13} + 840q^{14} + 2440q^{15} + 1536q^{16} + 1220q^{17} + 2996q^{18} + 1897q^{19} + 976q^{20} - 1198q^{21} + 2884q^{22} + 7240q^{23} + 1088q^{24} + O(q^{25})$

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

$x^{6} - 78x^{5} - 12971x^{4} + 1153300x^{3} - 6315744x^{2} - 121153904x - 267365904$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
43	$+1$

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
$\operatorname{tr} a_p$	24	17	61	210	721	1987	1220	1897	7240	−3869	1538	−4095	3632	−11094	12203	−38503	−59836	32454	−55529	−61114	−75018	−34585	99211	55834	116904

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 43, 86]: [1, 2, 20, 57] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 4 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(86, 6)[3]   # 86.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 86:6 1000