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

Newform orbit 186.4.a.d

$q$ -expansion

$q - 2q^{2} + 3q^{3} + 4q^{4} + 15q^{5} - 6q^{6} + 17q^{7} - 8q^{8} + 9q^{9} - 30q^{10} + 24q^{11} + 12q^{12} + 2q^{13} - 34q^{14} + 45q^{15} + 16q^{16} - 48q^{17} - 18q^{18} - 115q^{19} + 60q^{20} + 51q^{21} - 48q^{22} + 30q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

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

$x - 66$

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 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
$\operatorname{tr} a_p$	−2	3	15	17	24	2	−48	−115	30	264	31	−160	−51	128	480	132	309	−280	−604	−159	−652	−838	−690	−534	329

$\operatorname{tr} a_n$ 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)[3]   # 186.4.a.d

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

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