Atlas › Modular forms › Weight 6 › Level 56 › 56.6.a › 56.6.a.e

Newform orbit 56.6.a.e

Trace form

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

$2q - 6q^{3} + 82q^{5} - 98q^{7} + 222q^{9} + 340q^{11} + 910q^{13} + 1824q^{15} + 3216q^{17} - 674q^{19} + 294q^{21} - 1104q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{345})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 76x - 4076$

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	−6	82	−98	340	910	3216	−674	−1104	8064	−6212	−8512	−1304	−10004	−12748	−11220	−12018	102738	24136	89720	−55588	48824	35782	−18300	−69984

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 2, 4, 5, 7, 12, 23, 44] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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, 6)[4]   # 56.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 56:6 1000