Atlas › Modular forms › Weight 4 › Level 184 › 184.4.a › 184.4.a.a

Newform orbit 184.4.a.a

$q$ -expansion

$q - 4q^{3} + 22q^{5} + 8q^{7} - 11q^{9} - 20q^{11} + 22q^{13} - 88q^{15} + 98q^{17} - 12q^{19} - 32q^{21} + 23q^{23} + O(q^{25})$

Coefficient field

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

$x - 18$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 184$ )
23	$-1$

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	−4	22	8	−20	22	98	−12	23	−10	192	−106	186	332	−544	390	−716	110	836	−280	−486	288	180	650	−1262

$\operatorname{tr} a_n$ 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 184, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.134 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, 4, 8, 23, 46, 92, 184]: [1, 2, 3, 5, 7, 20, 39, 76] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 16 = 1 x 16 (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(184, 4)[0]   # 184.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 184:4 1000