Atlas › Modular forms › Weight 2 › Level 568 › 568.2.a › 568.2.a.d

Newform orbit 568.2.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 5q^{3} - q^{5} + 9q^{9} + 8q^{11} - 4q^{13} + 6q^{15} + 10q^{17} + 11q^{19} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{4} - 2x^{3} - 139x^{2} - 103x + 1550$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 568$ )
71	$-1$

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	5	−1	0	8	−4	10	11	2	−7	0	−5	0	13	10	−2	10	−26	6	4	−3	−7	23	23	−14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 8, 71, 142, 284, 568]: [0, 1, 2, 3, 7, 20, 39, 76] (certified at ell = 2147483171)
T = [(3, 1), (5, 3)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 3 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(568, 2)[3]   # 568.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 568:2 1000