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

Newform orbit 184.4.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 5q^{3} - 2q^{5} - 32q^{7} + 79q^{9} + 56q^{11} + 139q^{13} + 230q^{15} - 6q^{17} + 108q^{21} + 92q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} - 3x^{3} - 649x^{2} - 1481x + 46740$

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	5	−2	−32	56	139	−6	0	92	489	−127	826	91	−762	397	−430	340	714	−1832	−1047	−27	−794	−502	6	−1970

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

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)[5]   # 184.4.a.f

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

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