Atlas › Modular forms › Weight 4 › Level 186 › 186.4.a › 186.4.a.e

Newform orbit 186.4.a.e

$q$ -expansion

$q + 2q^{2} - 3q^{3} + 4q^{4} - 7q^{5} - 6q^{6} - 3q^{7} + 8q^{8} + 9q^{9} - 14q^{10} - 18q^{11} - 12q^{12} - 52q^{13} - 6q^{14} + 21q^{15} + 16q^{16} - 60q^{17} + 18q^{18} - 119q^{19} - 28q^{20} + 9q^{21} - 36q^{22} - 20q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x + 16$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
31	$+1$

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	−3	−7	−3	−18	−52	−60	−119	−20	178	−31	58	−285	230	164	180	351	288	324	−65	−386	−758	814	594	−1615

$\operatorname{tr} a_n$ 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 186, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 3, 6, 31, 62, 93, 186]: [1, 2, 2, 5, 9, 26, 34, 100] (certified at ell = 2147482951)
T = [(5, 1), (7, 3)]; h of degree 14 = 1 x 14 (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(186, 4)[4]   # 186.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 186:4 1000