Atlas › Modular forms › Weight 4 › Level 216 › 216.4.a › 216.4.a.c

Newform orbit 216.4.a.c

$q$ -expansion

$q + q^{5} - 9q^{7} - 17q^{11} - 44q^{13} + 56q^{17} - 94q^{19} - 50q^{23} + O(q^{25})$

Coefficient field

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

$x + 26$

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 $a_p = -p^{k/2-1} w_p$ .

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	−9	−17	−44	56	−94	−50	−30	−139	−174	318	−242	−630	547	−236	328	614	296	433	−56	−1225	1506	1391

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{3/2}$ 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 0.202 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 engine's checks

dim M^+ at levels [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216]: [1, 2, 2, 3, 5, 5, 4, 9, 11, 16, 10, 21, 29, 40, 57, 112] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 12 = 1 x 12 (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(216, 4)[2]   # 216.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 216:4 1000