Atlas › Modular forms › Weight 2 › Level 539 › 539.2.a › 539.2.a.f

Newform orbit 539.2.a.f

Trace form

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

$2q - 2q^{3} + 6q^{4} + 4q^{5} + 10q^{6} + 6q^{9} - 2q^{11} - 6q^{12} - 2q^{13} - 4q^{15} - 2q^{16} + 2q^{17} - 20q^{18} - 4q^{19} + 12q^{20} - 4q^{23} + 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{13}$ , whose minimal polynomial is

$x^{2} - 45$

Atkin–Lehner signs

$p$	$w_p$
7	— ( $7^2 \mid 539$ )
11	$+1$

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	4	0	−2	−2	2	−4	−4	8	10	−8	18	16	−10	8	−2	10	20	−12	6	0	−4	−4	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.338 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, 7, 11, 49, 77, 539]: [0, 1, 2, 5, 10, 58] (certified at ell = 2147482681)
T = [(2, 1), (3, 3), (5, 5), (13, 7)]; h of degree 34 = 1 x 34 (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(539, 2)[5]   # 539.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 539:2 1000