Atlas › Modular forms › Weight 2 › Level 572 › 572.2.a › 572.2.a.e

Newform orbit 572.2.a.e

Trace form

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

$3q - 2q^{3} - q^{5} - 7q^{7} + 3q^{9} + 3q^{11} - 3q^{13} - 13q^{15} - 6q^{17} - 3q^{19} - 10q^{23} + O(q^{25})$

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

$x^{3} + 3x^{2} - x - 4$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 572$ )
11	$-1$
13	$+1$

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	−2	−1	−7	3	−3	−6	−3	−10	−14	−23	−11	−1	−14	8	5	−1	4	−9	7	3	4	5	25	−1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 11, 13, 22, 26, 44, 52, 143, 286, 572]: [0, 1, 2, 2, 1, 5, 5, 9, 10, 16, 46, 90] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (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(572, 2)[4]   # 572.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 572:2 1000