Atlas › Modular forms › Weight 2 › Level 432 › 432.2.a › 432.2.a.b

Newform orbit 432.2.a.b

$q$ -expansion

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

Coefficient field

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

$x - 290$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 432$ )
3	— ( $3^2 \mid 432$ )

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 432.b (Cremona's 432f). Its optimal curve 432f1 is

$y^2 = x^3 + 21x + 26$

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	−3	1	−3	−4	0	−2	−6	−6	−5	2	6	10	6	−9	12	8	−14	0	−7	−8	−3	18	−1

$\operatorname{tr} a_n$ 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 432, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.126 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, 36, 48, 54, 72, 108, 144, 216, 432]: [0, 1, 1, 2, 3, 3, 2, 5, 4, 5, 8, 4, 9, 12, 11, 16, 21, 28, 40, 76] (certified at ell = 2147483629)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12), (47, 13), (53, 14), (59, 15), (61, 16)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(432, 2)[1]   # 432.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 432:2 1000