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

Newform orbit 86.6.a.c

Trace form

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

$5q - 20q^{2} - q^{3} + 80q^{4} + 61q^{5} + 4q^{6} + 122q^{7} - 320q^{8} + 184q^{9} - 244q^{10} - 77q^{11} - 16q^{12} - 67q^{13} - 488q^{14} - 1936q^{15} + 1280q^{16} - 294q^{17} - 736q^{18} + 4811q^{19} + 976q^{20} + 7730q^{21} + 308q^{22} + 7586q^{23} + 64q^{24} + O(q^{25})$

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

$x^{5} - 60x^{4} - 10011x^{3} + 713162x^{2} + 8135412x - 893359800$

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
$\operatorname{tr} a_p$	−20	−1	61	122	−77	−67	−294	4811	7586	14831	18076	16441	14446	9245	3207	−50377	−70116	54084	23345	−43522	−41244	28419	−55883	35280	242558

$\operatorname{tr} a_n$ 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)[2]   # 86.6.a.c

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

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