Atlas › Modular forms › Weight 2 › Level 528 › 528.2.a › 528.2.a.j

Newform orbit 528.2.a.j

$q$ -expansion

$q + q^{3} + 2q^{5} + 4q^{7} + q^{9} + q^{11} - 6q^{13} + 2q^{15} + 2q^{17} - 4q^{19} + 4q^{21} - 4q^{23} + O(q^{25})$

Coefficient field

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

$x + 316$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 528$ )
3	$-1$
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 528.j (Cremona's 528j). Its optimal curve 528j1 is

$y^2 = x^3 + x^2 - 32x - 12$

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
$\operatorname{tr} a_p$	0	1	2	4	1	−6	2	−4	−4	6	0	6	−6	−4	12	2	−12	−14	−4	12	−6	4	−4	10	−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 528, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.665 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, 4, 6, 8, 11, 12, 16, 22, 24, 33, 44, 48, 66, 88, 132, 176, 264, 528]: [0, 1, 1, 2, 3, 3, 2, 5, 4, 5, 8, 6, 9, 12, 16, 16, 30, 28, 56, 104] (certified at ell = 2147482921)
T = [(5, 1), (7, 2), (13, 3), (19, 4), (23, 5), (29, 6), (17, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 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(528, 2)[9]   # 528.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 528:2 1000