Atlas › Modular forms › Weight 2 › Level 490 › 490.2.a › 490.2.a.g

Newform orbit 490.2.a.g

$q$ -expansion

$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + q^{8} + q^{9} - q^{10} + 3q^{11} - 2q^{12} + 5q^{13} + 2q^{15} + q^{16} + 6q^{17} + q^{18} - q^{19} - q^{20} + 3q^{22} + 3q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 2T_{11} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{13} + 7T_{17} + 8T_{31}$ is a root of

$x - 23$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$+1$
7	— ( $7^2 \mid 490$ )

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 490.g (Cremona's 490e). Its optimal curve 490e1 is

$y^2 + xy = x^3 - x - 15$

of rank 0 with torsion of order 3. 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	−1	0	3	5	6	−1	3	−6	−4	11	3	−10	3	3	0	−4	−4	12	−4	−10	−12	6	14

$\operatorname{tr} a_n$ 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 490, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.449 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, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490]: [0, 1, 1, 1, 3, 4, 6, 5, 16, 16, 30, 88] (certified at ell = 2147483353)
T = [(3, 1), (11, 2), (19, 3), (23, 4), (29, 5), (13, 6), (17, 7), (31, 8)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 5 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(490, 2)[6]   # 490.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 490:2 1000