Atlas › Modular forms › Weight 2 › Level 650 › 650.2.a › 650.2.a.j

Newform orbit 650.2.a.j

$q$ -expansion

$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} - 2q^{9} + 6q^{11} - q^{12} - q^{13} + q^{14} + q^{16} + 3q^{17} - 2q^{18} + 2q^{19} - q^{21} + 6q^{22} - q^{24} + O(q^{25})$

Coefficient field

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

$x - 51$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	— ( $5^2 \mid 650$ )
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 650.j (Cremona's 650h). Its optimal curve 650h1 is

$y^2 + xy + y = x^3 + x^2 + 12x + 31$

of rank 0 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
$\operatorname{tr} a_p$	1	−1	0	1	6	−1	3	2	0	6	−4	7	0	1	−3	0	−6	8	−14	−3	−2	8	−12	−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 650, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.748 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, 10, 13, 25, 26, 50, 65, 130, 325, 650]: [0, 1, 1, 3, 1, 3, 5, 9, 8, 24, 36, 108] (certified at ell = 2147482921)
T = [(3, 1), (7, 2), (11, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 19 = 1 x 19 (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(650, 2)[9]   # 650.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 650:2 1000