Atlas › Modular forms › Weight 2 › Level 990 › 990.2.a › 990.2.a.k

Newform orbit 990.2.a.k

$q$ -expansion

$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} + q^{11} + q^{16} + 2q^{17} + 2q^{19} + q^{20} + q^{22} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{7} + 2T_{13} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{31} + 8T_{37}$ is a root of

$x - 8$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	— ( $3^2 \mid 990$ )
5	$-1$
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 990.k (Cremona's 990i). Its optimal curve 990i1 is

$y^2 + xy + y = x^3 - x^2 - 137x - 539$

of rank 0 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	1	0	2	2	6	2	0	−6	2	−2	−2	2	8	4	−4	−2	−10	4	12	0	−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 990, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.634 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, 5, 6, 9, 10, 11, 15, 18, 22, 30, 33, 45, 55, 66, 90, 99, 110, 165, 198, 330, 495, 990]: [0, 1, 1, 1, 3, 2, 3, 2, 4, 5, 5, 10, 6, 8, 8, 16, 22, 14, 22, 28, 40, 80, 76, 224] (certified at ell = 2147482921)
T = [(7, 1), (13, 2), (17, 3), (19, 4), (23, 5), (29, 6), (31, 7), (37, 8)]; h of degree 14 = 1 x 14 (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(990, 2)[10]   # 990.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 990:2 1000