Atlas › Modular forms › Weight 2 › Level 864 › 864.2.a › 864.2.a.d

Newform orbit 864.2.a.d

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 7159T_{5} + 2426T_{7} + 9370T_{13} + 5254T_{17} + 3871T_{19} + 2211T_{11} + 9392T_{23} + 2182T_{29}$ is a root of

$x + 88417$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 864$ )
3	— ( $3^2 \mid 864$ )

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 864.d (Cremona's 864l). Its optimal curve 864l1 is

$y^2 = x^3 - 216x + 1296$

of rank 1 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
$\operatorname{tr} a_p$	0	0	−2	3	−6	−3	2	−3	−6	8	0	7	−8	−12	−6	−4	−6	−1	−3	−12	−15	9	12	10	9

$\operatorname{tr} a_n$ 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 864, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.161 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, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 96, 108, 144, 216, 288, 432, 864]: [0, 1, 1, 2, 3, 3, 2, 5, 4, 5, 8, 4, 6, 9, 12, 11, 16, 20, 21, 28, 40, 52, 76, 148] (certified at ell = 2147482873)
T = [(5, 7159), (7, 2426), (13, 9370), (17, 5254), (19, 3871), (11, 2211), (23, 9392), (29, 2182)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 10 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(864, 2)[3]   # 864.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 864:2 1000