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

Newform orbit 187.4.a.f

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 10q^{2} + 3q^{3} + 60q^{4} + 29q^{5} - 43q^{6} + 39q^{7} + 120q^{8} + 119q^{9} + 53q^{10} + 132q^{11} - 27q^{12} + 171q^{13} - 3q^{14} + 6q^{15} + 356q^{16} + 204q^{17} + 365q^{18} - 3q^{19} + 307q^{20} + 352q^{21} + 110q^{22} + 268q^{23} - 53q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 13x^{11} - 172x^{10} + 2291x^{9} + 10131x^{8} - 128357x^{7} - 280591x^{6} + 2604463x^{5} + 4391790x^{4} - 19719558x^{3} - 29372545x^{2} + 49234486x + 67321498$

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 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
$\operatorname{tr} a_p$	10	3	29	39	132	171	204	−3	268	466	98	201	221	707	−79	932	−72	−1012	−598	698	195	1299	3533	1691	−1178

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.131 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)[5]   # 187.4.a.f

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

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