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

Newform orbit 539.2.a.d

$q$ -expansion

$q + q^{2} - 2q^{3} - q^{4} + 2q^{5} - 2q^{6} - 3q^{8} + q^{9} + 2q^{10} + q^{11} + 2q^{12} - 4q^{13} - 4q^{15} - q^{16} - 4q^{17} + q^{18} - 2q^{20} + q^{22} - 4q^{23} + 6q^{24} + O(q^{25})$

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

$x + 23$

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.d (Cremona's 539c). Its optimal curve 539c1 is

$y^2 + xy + y = x^3 + 170x - 3237$

of rank 1 with torsion of order 2. 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
$\operatorname{tr} a_p$	1	−2	2	0	1	−4	−4	0	−4	−6	−10	−6	−4	12	10	−6	−2	0	8	−12	8	8	0	6	10

$\operatorname{tr} a_n$ 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)[3]   # 539.2.a.d

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

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