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

Newform orbit 187.4.a.e

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 4q^{2} + 11q^{3} + 40q^{4} + 47q^{5} + 105q^{6} - 17q^{7} + 84q^{8} + 107q^{9} - 7q^{10} - 110q^{11} + 101q^{12} + 15q^{13} + 45q^{14} + 224q^{15} + 140q^{16} - 170q^{17} + 81q^{18} + q^{19} + 289q^{20} + 16q^{21} - 44q^{22} + 340q^{23} + 663q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 15x^{9} - 241x^{8} + 3856x^{7} + 17756x^{6} - 301625x^{5} - 714331x^{4} + 9581074x^{3} + 18879033x^{2} - 105713102x - 220475646$

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
$\operatorname{tr} a_p$	4	11	47	−17	−110	15	−170	1	340	452	298	619	367	−385	365	1624	−414	−1162	−1216	850	−983	−1165	−1391	2225	700

$\operatorname{tr} a_n$ 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)[4]   # 187.4.a.e

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

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