Atlas › Modular forms › Weight 2 › Level 630 › 630.2.a › 630.2.a.b

Newform orbit 630.2.a.b

$q$ -expansion

$q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} - 4q^{11} - 2q^{13} - q^{14} + q^{16} - 2q^{17} - 4q^{19} - q^{20} + 4q^{22} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{11} + 2T_{13} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{17} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53} + 13T_{59} + 14T_{61} + 15T_{67} + 16T_{71}$ is a root of

$x + 252$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	— ( $3^2 \mid 630$ )
5	$+1$
7	$-1$

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 630.b (Cremona's 630d). Its optimal curve 630d1 is

$y^2 + xy = x^3 - x^2 + 90x + 436$

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	1	−4	−2	−2	−4	8	−6	−8	−2	−2	−12	8	−6	−4	−2	12	−8	−14	0	−12	−2	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 630, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.349 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, 7, 9, 10, 14, 15, 18, 21, 30, 35, 42, 45, 63, 70, 90, 105, 126, 210, 315, 630]: [0, 1, 1, 1, 3, 1, 2, 3, 4, 4, 5, 4, 10, 6, 12, 8, 10, 16, 22, 20, 28, 56, 52, 152] (certified at ell = 2147483629)
T = [(11, 1), (13, 2), (19, 3), (23, 4), (29, 5), (17, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12), (59, 13), (61, 14), (67, 15), (71, 16)]; h of degree 10 = 1 x 10 (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(630, 2)[1]   # 630.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 630:2 1000