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

Newform orbit 86.4.a.b

$q$ -expansion

$q + 2q^{2} - 4q^{3} + 4q^{4} - 14q^{5} - 8q^{6} - 14q^{7} + 8q^{8} - 11q^{9} - 28q^{10} - 11q^{11} - 16q^{12} - 9q^{13} - 28q^{14} + 56q^{15} + 16q^{16} + 9q^{17} - 22q^{18} - 46q^{19} - 56q^{20} + 56q^{21} - 22q^{22} - 19q^{23} - 32q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 18$

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
$\operatorname{tr} a_p$	2	−4	−14	−14	−11	−9	9	−46	−19	216	−155	−76	5	−43	−392	579	−588	28	−621	−146	−192	−664	−1239	−1622	827

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 86, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[1]   # 86.4.a.b

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

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