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

Newform orbit 86.4.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} + 2q^{3} + 8q^{4} - 14q^{5} - 4q^{6} - 22q^{7} - 16q^{8} + 2q^{9} + 28q^{10} + 10q^{11} + 8q^{12} - 110q^{13} + 44q^{14} - 140q^{15} + 32q^{16} + 14q^{17} - 4q^{18} - 272q^{19} - 56q^{20} - 220q^{21} - 20q^{22} + 214q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 12x - 12$

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
$\operatorname{tr} a_p$	−4	2	−14	−22	10	−110	14	−272	214	−106	−194	236	−210	86	160	186	68	−994	590	−72	−1474	668	382	726	−326

$\operatorname{tr} a_n$ 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)[2]   # 86.4.a.c

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

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