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

Newform orbit 56.8.a.d

Trace form

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

$3q + 12q^{3} - 598q^{5} - 1029q^{7} + 4071q^{9} + 7972q^{11} + 9170q^{13} - 14496q^{15} + 43878q^{17} + 42156q^{19} - 4116q^{21} + 123712q^{23} + O(q^{25})$

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

$x^{3} + 586x^{2} + 38300x - 10484200$

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	12	−598	−1029	7972	9170	43878	42156	123712	−52158	205040	4186	−428482	−154700	353392	−3590550	−623356	−1181086	−2107940	3924424	1784062	2759968	5270364	4259070	11453542

$\operatorname{tr} a_n$ 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)[3]   # 56.8.a.d

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

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