Atlas › Modular forms › Weight 6 › Level 27 › 27.6.a › 27.6.a.b

Newform orbit 27.6.a.b

Trace form

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

$2q - 9q^{2} + 53q^{4} - 72q^{5} - 8q^{7} - 639q^{8} - 441q^{10} - 522q^{11} - 704q^{13} + 1413q^{14} + 3857q^{16} - 216q^{17} - 2840q^{19} + 4977q^{20} - 99q^{22} + 36q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 225x - 19512$

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	−72	−8	−522	−704	−216	−2840	36	−12240	−1064	9004	−5688	784	−1116	−4536	67320	−49904	−42176	43848	47218	−49616	102294	35856	169966

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.225 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]: [1, 3, 6, 16] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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, 6)[1]   # 27.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 27:6 1000