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

Newform orbit 81.6.a.e

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 150q^{4} + 372q^{7} + 630q^{10} + 930q^{13} + 3282q^{16} + 4548q^{19} + 14652q^{22} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} - 18306x^{4} + 87276609x^{2} - 108699367500$

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
$\operatorname{tr} a_p$	0	0	0	372	0	930	0	4548	0	0	12864	−258	0	−35916	0	0	0	68142	−15540	0	−32010	110532	0	0	357420

$\operatorname{tr} a_n$ 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)[4]   # 81.6.a.e

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

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