Atlas › Modular forms › Weight 2 › Level 540 › 540.2.a › 540.2.a.d

Newform orbit 540.2.a.d

$q$ -expansion

$q + q^{5} - 4q^{7} - 6q^{11} - 4q^{13} + 3q^{17} - 7q^{19} + 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 6146T_{7} + 2849T_{11} + 7198T_{13} + 8490T_{17} + 134T_{19} + 8011T_{23} + 7196T_{29} + 3588T_{31}$ is a root of

$x - 1045$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 540$ )
3	— ( $3^2 \mid 540$ )
5	$-1$

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 540.d (Cremona's 540b). Its optimal curve 540b1 is

$y^2 = x^3 + 3x + 1$

of rank 1 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	0	1	−4	−6	−4	3	−7	9	0	−7	2	−6	2	0	−9	12	−7	2	−6	2	−1	−9	6	8

$\operatorname{tr} a_n$ 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 540, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.21 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, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540]: [0, 1, 1, 2, 1, 3, 2, 3, 5, 4, 5, 6, 4, 10, 9, 8, 11, 18, 22, 21, 20, 42, 58, 114] (certified at ell = 2147483629)
T = [(7, 6146), (11, 2849), (13, 7198), (17, 8490), (19, 134), (23, 8011), (29, 7196), (31, 3588)]; h of degree 6 = 1 x 6 (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(540, 2)[3]   # 540.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 540:2 1000