Atlas › Modular forms › Weight 4 › Level 189 › 189.4.a › 189.4.a.f

Newform orbit 189.4.a.f

Trace form

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

$2q - 10q^{4} + 14q^{7} + 6q^{10} - 92q^{13} + 2q^{16} - 242q^{19} - 186q^{22} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 48$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 189$ )
7	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	0	0	14	0	−92	0	−242	0	0	−266	−374	0	376	0	0	0	−176	−572	0	−1016	2560	0	0	2392

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.154 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, 3, 7, 9, 21, 27, 63, 189]: [1, 2, 3, 4, 10, 10, 26, 74] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 24 = 1 x 24 (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(189, 4)[5]   # 189.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 189:4 1000