Atlas › Modular forms › Weight 4 › Level 190 › 190.4.a › 190.4.a.e

Newform orbit 190.4.a.e

Trace form

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

$2q - 4q^{2} + 2q^{3} + 8q^{4} - 10q^{5} - 4q^{6} + 8q^{7} - 16q^{8} + 2q^{9} + 20q^{10} + 28q^{11} + 8q^{12} + 2q^{13} - 16q^{14} - 10q^{15} + 32q^{16} + 152q^{17} - 4q^{18} - 38q^{19} - 40q^{20} - 136q^{21} - 56q^{22} + 284q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} - 26x - 1154$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$+1$
19	$+1$

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	2	−10	8	28	2	152	−38	284	144	128	−194	608	288	432	810	324	−1076	658	−1696	832	−1428	−508	576	834

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.17 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, 5, 10, 19, 38, 95, 190]: [1, 2, 3, 7, 6, 17, 32, 94] (certified at ell = 2147483629)
T = [(3, 1), (7, 3)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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(190, 4)[4]   # 190.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 190:4 1000