Atlas › Modular forms › Weight 2 › Level 560 › 560.2.a › 560.2.a.h

Newform orbit 560.2.a.h

Trace form

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

$2q + q^{3} - 2q^{5} + 2q^{7} + 11q^{9} - 7q^{11} + 3q^{13} - q^{15} + 5q^{17} - 2q^{19} + q^{21} - 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{33})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{11} + 3T_{13} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{17} + 8T_{31}$ , whose minimal polynomial is

$x^{2} - 123x + 3576$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 560$ )
5	$+1$
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	1	−2	2	−7	3	5	−2	−2	−3	16	−4	2	6	−3	10	−16	6	8	−16	−12	−13	−4	18	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.684 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, 5, 7, 8, 10, 14, 16, 20, 28, 35, 40, 56, 70, 80, 112, 140, 280, 560]: [0, 1, 2, 1, 1, 3, 3, 4, 4, 6, 7, 6, 10, 12, 16, 16, 20, 30, 56, 104] (certified at ell = 2147483629)
T = [(3, 1), (11, 2), (13, 3), (19, 4), (23, 5), (29, 6), (17, 7), (31, 8)]; h of degree 12 = 1 x 12 (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(560, 2)[7]   # 560.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 560:2 1000