Atlas › Modular forms › Weight 4 › Level 234 › 234.4.a › 234.4.a.f

Newform orbit 234.4.a.f

$q$ -expansion

$q - 2q^{2} + 4q^{4} + 20q^{5} - 32q^{7} - 8q^{8} - 40q^{10} - 50q^{11} - 13q^{13} + 64q^{14} + 16q^{16} + 30q^{17} - 120q^{19} + 80q^{20} + 100q^{22} + 20q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{17}$ is a root of

$x + 116$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	— ( $3^2 \mid 234$ )
13	$+1$

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
$\operatorname{tr} a_p$	−2	0	20	−32	−50	−13	30	−120	20	−82	−44	−306	−108	−356	178	−198	−94	−62	−140	778	62	−1096	462	−1224	614

$\operatorname{tr} a_n$ 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 234, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.366 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, 6, 9, 13, 18, 26, 39, 78, 117, 234]: [1, 2, 2, 5, 4, 5, 11, 13, 16, 46, 44, 130] (certified at ell = 2147483629)
T = [(5, 1), (7, 3), (11, 5), (17, 7)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 6 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(234, 4)[5]   # 234.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 234:4 1000