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

Newform orbit 187.4.a.c

Trace form

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

$4q + 2q^{2} - 6q^{3} + 24q^{4} - 10q^{5} - 18q^{6} + 16q^{7} - 84q^{8} - 64q^{9} - 50q^{10} - 44q^{11} - 16q^{12} - 10q^{13} - 92q^{14} - 40q^{15} + 136q^{16} + 68q^{17} + 88q^{18} - 114q^{19} - 120q^{20} - 44q^{21} - 22q^{22} - 258q^{23} - 84q^{24} + O(q^{25})$

Coefficient field

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

$x^{4} + 4x^{3} - 24x^{2} - 116x - 109$

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
$\operatorname{tr} a_p$	2	−6	−10	16	−44	−10	68	−114	−258	−10	14	126	−868	−336	−208	636	−412	128	124	−682	422	−598	−2368	−1484	−990

$\operatorname{tr} a_n$ 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)[2]   # 187.4.a.c

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

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