Atlas › Modular forms › Weight 2 › Level 187 › 187.2.a › 187.2.a.f

Newform orbit 187.2.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + q^{2} + q^{3} + 5q^{4} + 3q^{5} - 4q^{6} + 9q^{8} + 11q^{9} - 12q^{10} - 4q^{11} - 2q^{13} + 5q^{15} + 19q^{16} + 4q^{17} - 7q^{18} - 2q^{19} - 6q^{20} - q^{22} + 5q^{23} - 26q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} - 4x^{3} - 90x^{2} + 344x + 173$

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
$\operatorname{tr} a_p$	1	1	3	0	−4	−2	4	−2	5	12	−17	19	6	−4	4	28	15	12	−1	17	−6	−22	8	−13	17

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.174 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 11, 17, 187]: [0, 2, 2, 20] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 2 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, 2)[5]   # 187.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 187:2 1000