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

Newform orbit 86.6.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 12q^{2} + 8q^{3} + 48q^{4} - 14q^{5} - 32q^{6} - 74q^{7} - 192q^{8} - 221q^{9} + 56q^{10} + 353q^{11} + 128q^{12} + 117q^{13} + 296q^{14} - 268q^{15} + 768q^{16} - 101q^{17} + 884q^{18} - 2774q^{19} - 224q^{20} - 1708q^{21} - 1412q^{22} - 6487q^{23} - 512q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 6x^{2} - 932x - 7128$

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 odd 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
$\operatorname{tr} a_p$	−12	8	−14	−74	353	117	−101	−2774	−6487	−5924	−15823	−10212	−2981	−5547	34612	6901	29840	−2080	−25937	22526	−2804	−3500	35761	31442	−134511

$\operatorname{tr} a_n$ 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)[0]   # 86.6.a.a

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

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