Atlas › Modular forms › Weight 2 › Level 529 › 529.2.a › 529.2.a.e

Newform orbit 529.2.a.e

Trace form

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

$2q + 2q^{2} + 2q^{4} + 2q^{5} + 4q^{6} + 4q^{7} + 6q^{8} - 2q^{9} - 6q^{10} + 8q^{12} - 6q^{13} + 8q^{14} - 8q^{15} + 6q^{16} - 2q^{18} + 4q^{19} - 14q^{20} + 4q^{21} - 4q^{22} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{2} - 40x + 272$

Atkin–Lehner signs

$p$	$w_p$
23	— ( $23^2 \mid 529$ )

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
$\operatorname{tr} a_p$	2	0	2	4	0	−6	0	4	0	6	12	−8	14	12	0	14	4	2	−8	−4	−2	−8	−20	6	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 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, 23, 529]: [0, 3, 47] (certified at ell = 2147474627)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 8 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(529, 2)[4]   # 529.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 529:2 1000