Atlas › Modular forms › Weight 12 › Level 27 › 27.12.a › 27.12.a.e

Newform orbit 27.12.a.e

Trace form

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

$4q + 9q^{2} + 6109q^{4} + 12060q^{5} - 12076q^{7} + 311355q^{8} + 217035q^{10} - 526788q^{11} - 2227432q^{13} - 1040157q^{14} + 9350785q^{16} + 8884296q^{17} + 6172784q^{19} + 80749755q^{20} - 78178023q^{22} + 76901040q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 12069x^{3} - 75287475x^{2} + 803711510325x + 857408118709122$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 27$ )

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
$\operatorname{tr} a_p$	9	0	12060	−12076	−526788	−2227432	8884296	6172784	76901040	180046080	−35694652	59658848	−540559152	−2521421368	−2691337608	−132122340	−8139028320	−3516171856	−711923512	36309366072	28769206364	−108210339448	68197592556	69699290616	−153951848548

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.546 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, 9, 27]: [2, 5, 12, 34] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 7 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(27, 12)[4]   # 27.12.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 27:12 1000