Atlas › Modular forms › Weight 4 › Level 187 › 187.4.a › 187.4.a.a

Newform orbit 187.4.a.a

$q$ -expansion

$q + q^{2} + 4q^{3} - 7q^{4} + 6q^{5} + 4q^{6} - 24q^{7} - 15q^{8} - 11q^{9} + 6q^{10} - 11q^{11} - 28q^{12} - 58q^{13} - 24q^{14} + 24q^{15} + 41q^{16} + 17q^{17} - 11q^{18} + 28q^{19} - 42q^{20} - 96q^{21} - 11q^{22} - 24q^{23} - 60q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x - 5$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
17	$-1$

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
$\operatorname{tr} a_p$	1	4	6	−24	−11	−58	17	28	−24	222	−112	−394	410	−204	240	−386	564	−530	108	−392	−406	1008	236	1354	−1774

$\operatorname{tr} a_n$ 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 187, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.131 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, 11, 17, 187]: [1, 4, 6, 56] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 6 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(187, 4)[0]   # 187.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 187:4 1000