Atlas › Modular forms › Weight 4 › Level 86 › 86.4.a › 86.4.a.e

Newform orbit 86.4.a.e

Trace form

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

$4q + 8q^{2} + 11q^{3} + 16q^{4} + q^{5} + 22q^{6} + 42q^{7} + 32q^{8} + 79q^{9} + 2q^{10} + 34q^{11} + 44q^{12} - 42q^{13} + 84q^{14} - 160q^{15} + 64q^{16} - 3q^{17} + 158q^{18} + 27q^{19} + 4q^{20} + 2q^{21} + 68q^{22} - 301q^{23} + 88q^{24} + O(q^{25})$

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

$x^{4} - 12x^{3} - 153x^{2} + 1364x + 2364$

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
$\operatorname{tr} a_p$	8	11	1	42	34	−42	−3	27	−301	−235	−7	−403	361	172	285	−400	456	−74	978	992	1030	1621	1420	288	1311

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 12, 35] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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, 4)[4]   # 86.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 86:4 1000