Atlas › Modular forms › Weight 8 › Level 27 › 27.8.a › 27.8.a.e

Newform orbit 27.8.a.e

Trace form

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

$2q + 9q^{2} + 77q^{4} + 180q^{5} + 700q^{7} + 1827q^{8} + 1395q^{10} + 10890q^{11} - 5480q^{13} + 29475q^{14} - 15967q^{16} + 16416q^{17} + 16024q^{19} + 12195q^{20} + 60705q^{22} + 24372q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{65})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 189x + 7614$

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	180	700	10890	−5480	16416	16024	24372	−143280	−38708	455620	731880	−1088840	1561500	2610468	1731960	−620192	346600	−4242240	−3145190	10110616	644202	6021000	4098670

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 8, 22] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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, 8)[4]   # 27.8.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 27:8 1000