Atlas › Modular forms › Weight 8 › Level 56 › 56.8.a › 56.8.a.e

Newform orbit 56.8.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 28q^{3} + 138q^{5} + 1029q^{7} - 649q^{9} + 2052q^{11} + 10050q^{13} + 7568q^{15} + 33942q^{17} + 39276q^{19} + 9604q^{21} + 64128q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{3} - 166x^{2} - 214708x + 46620664$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 56$ )
7	$-1$

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	28	138	1029	2052	10050	33942	39276	64128	255522	146880	820794	744798	−523500	−1098432	−186870	−3388476	−1358526	−4556100	−4868616	2207118	−1301712	549996	−4950450	2448438

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.313 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, 2, 4, 7, 8, 14, 28, 56]: [1, 3, 5, 5, 9, 16, 31, 60] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 4 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(56, 8)[4]   # 56.8.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 56:8 1000