Atlas › Modular forms › Weight 6 › Level 81 › 81.6.a › 81.6.a.a

Newform orbit 81.6.a.a

Trace form

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

$2q - 3q^{2} + 5q^{4} + 30q^{5} - 128q^{7} - 105q^{8} - 303q^{10} + 312q^{11} - 230q^{13} + 1740q^{14} - 1615q^{16} - 3150q^{17} - 2336q^{19} + 849q^{20} - 4080q^{22} + 4440q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{129})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 27x - 108$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 81$ )

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
$\operatorname{tr} a_p$	−3	0	30	−128	312	−230	−3150	−2336	4440	726	−8600	−3182	−14100	−10856	7944	45684	−66768	14218	−10400	24768	4774	−44264	30264	−49014	−194780

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 81]: [1, 3, 6, 16, 46] (certified at ell = 2147483179)
T = [(2, 1), (5, 1)]; 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(81, 6)[0]   # 81.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 81:6 1000