Atlas › Modular forms › Weight 2 › Level 585 › 585.2.a › 585.2.a.g

Newform orbit 585.2.a.g

$q$ -expansion

$q + q^{2} - q^{4} - q^{5} - 3q^{8} - q^{10} - 4q^{11} + q^{13} - q^{16} - 2q^{17} - 4q^{19} + q^{20} - 4q^{22} - 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ is a root of

$x + 137$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 585$ )
5	$+1$
13	$-1$

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 585.g (Cremona's 585f). Its optimal curve 585f1 is

$y^2 + xy = x^3 - x^2 - 990x - 11745$

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	0	−1	0	−4	1	−2	−4	−8	2	−8	6	6	−4	8	−6	12	−2	−4	0	−6	16	4	−10	18

$\operatorname{tr} a_n$ 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 585, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.586 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, 3, 5, 9, 13, 15, 39, 45, 65, 117, 195, 585]: [0, 1, 1, 2, 1, 4, 6, 8, 8, 16, 32, 88] (certified at ell = 2147483629)
T = [(2, 1), (7, 2), (11, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 6 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(585, 2)[6]   # 585.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 585:2 1000