Atlas › Modular forms › Weight 4 › Level 186 › 186.4.a › 186.4.a.i

Newform orbit 186.4.a.i

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{2} + 6q^{3} + 8q^{4} + 15q^{5} + 12q^{6} + 29q^{7} + 16q^{8} + 18q^{9} + 30q^{10} - 14q^{11} + 24q^{12} - 78q^{13} + 58q^{14} + 45q^{15} + 32q^{16} + 10q^{17} + 36q^{18} + 53q^{19} + 60q^{20} + 87q^{21} - 28q^{22} + 124q^{23} + 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{521})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 102x + 2080$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
31	$+1$

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	6	15	29	−14	−78	10	53	124	−84	−62	120	29	−526	400	−842	279	−378	−1376	−221	−328	−382	−494	−184	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 2, 3, 6, 31, 62, 93, 186]: [1, 2, 2, 5, 9, 26, 34, 100] (certified at ell = 2147482951)
T = [(5, 1), (7, 3)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(186, 4)[8]   # 186.4.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 186:4 1000