Atlas › Modular forms › Weight 2 › Level 216 › 216.2.a › 216.2.a.c

Newform orbit 216.2.a.c

$q$ -expansion

$q + q^{5} + 3q^{7} - 5q^{11} + 4q^{13} + 8q^{17} + 2q^{19} - 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x - 10$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 216$ )
3	— ( $3^2 \mid 216$ )

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 216.c (Cremona's 216c). Its optimal curve 216c1 is

$y^2 = x^3 - 27x + 918$

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
$\operatorname{tr} a_p$	0	0	1	3	−5	4	8	2	−2	−6	−7	−6	6	−2	−6	−5	4	−8	−10	8	1	16	11	−6	−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 216, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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, 18, 24, 27, 36, 54, 72, 108, 216]: [0, 1, 1, 2, 3, 3, 2, 5, 5, 8, 4, 9, 11, 16, 21, 40] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(216, 2)[2]   # 216.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 216:2 1000