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

Newform orbit 56.8.a.c

Trace form

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

$2q - 42q^{3} + 14q^{5} + 686q^{7} + 990q^{9} - 540q^{11} - 462q^{13} - 16728q^{15} - 9128q^{17} - 52654q^{19} - 14406q^{21} - 115864q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{249})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 28x - 15740$

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	−42	14	686	−540	−462	−9128	−52654	−115864	−68680	−235228	−87480	215152	−350372	−773668	594956	806274	3168830	−2815488	−754824	3907820	−5878360	2348346	−8096396	−18166120

$\operatorname{tr} a_n$ 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)[2]   # 56.8.a.c

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

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