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

Newform orbit 186.4.a.h

Trace form

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

$2q - 4q^{2} - 6q^{3} + 8q^{4} - 2q^{5} + 12q^{6} + 7q^{7} - 16q^{8} + 18q^{9} + 4q^{10} - 29q^{11} - 24q^{12} - 83q^{13} - 14q^{14} + 6q^{15} + 32q^{16} + 13q^{17} - 36q^{18} + 266q^{19} - 8q^{20} - 21q^{21} + 58q^{22} - 200q^{23} + 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{97})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 19x - 8664$

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	−2	7	−29	−83	13	266	−200	10	−62	184	29	616	587	−1278	157	661	771	−1198	−948	173	265	312	−2482

$\operatorname{tr} a_n$ 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)[7]   # 186.4.a.h

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

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