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

Newform orbit 187.4.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 9q^{2} + q^{3} + 3q^{4} - 19q^{5} - 3q^{6} + 9q^{7} + 63q^{8} + 60q^{9} + 57q^{10} - 33q^{11} + q^{12} - 11q^{13} - 27q^{14} + 36q^{15} - 213q^{16} + 51q^{17} - 180q^{18} - 71q^{19} - 19q^{20} - 180q^{21} + 99q^{22} - 64q^{23} + 21q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 8x^{2} - 49x - 95$

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
$\operatorname{tr} a_p$	−9	1	−19	9	−33	−11	51	−71	−64	−512	350	−251	−267	145	−961	−1004	−840	1642	−902	974	125	1201	939	61	728

$\operatorname{tr} a_n$ 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)[1]   # 187.4.a.b

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

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