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

Newform orbit 539.2.a.b

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{13}$ is a root of

$x - 10$

Atkin–Lehner signs

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

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Elliptic curve

By modularity this newform is the one attached to the isogeny class 539.b (Cremona's 539a). Its optimal curve 539a1 is

$y^2 + y = x^3 - x^2 - 4377x - 110013$

of rank 0 with torsion of order 1. Its $a_p$ , computed by Sagebrush's point-counting engine, equal the newform's for every prime $p < 1000$ .

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	−3	0	−1	4	6	−2	3	−6	−5	11	−6	8	0	−6	9	10	5	9	−2	−10	−12	3	1

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

Sato–Tate

The normalized $a_p / 2p^{1/2}$ for the primes $p < 1000$ not dividing 539, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[1]   # 539.2.a.b

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

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