Atlas › Modular forms › Weight 4 › Level 192 › 192.4.a › 192.4.a.k

Newform orbit 192.4.a.k

$q$ -expansion

$q + 3q^{3} + 14q^{5} + 36q^{7} + 9q^{9} - 36q^{11} - 54q^{13} + 42q^{15} - 22q^{17} + 36q^{19} + 108q^{21} + 144q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ is a root of

$x - 22$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 192$ )
3	$-1$

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
$\operatorname{tr} a_p$	0	3	14	36	−36	−54	−22	36	144	−50	108	−214	−446	252	−72	22	−684	466	−180	−576	−54	972	−684	346	−1134

$\operatorname{tr} a_n$ 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 192, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.32 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, 12, 16, 24, 32, 48, 64, 96, 192]: [1, 2, 2, 3, 5, 5, 9, 8, 16, 14, 28, 26, 52, 100] (certified at ell = 2147483489)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 5 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(192, 4)[10]   # 192.4.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 192:4 1000