Atlas › Modular forms › Weight 4 › Level 228 › 228.4.a › 228.4.a.e

Newform orbit 228.4.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 9q^{3} + 31q^{5} + 9q^{7} + 27q^{9} + 43q^{11} + 60q^{13} + 93q^{15} + 99q^{17} - 57q^{19} + 27q^{21} - 14q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + T_{7}$ , whose minimal polynomial is

$x^{3} - 40x^{2} - 580x + 26080$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 228$ )
3	$-1$
19	$+1$

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	9	31	9	43	60	99	−57	−14	218	192	522	80	141	−329	−578	−552	231	−204	−1100	933	−234	−1764	−570	1158

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.176 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, 12, 19, 38, 57, 76, 114, 228]: [1, 2, 2, 3, 5, 9, 6, 17, 22, 33, 64, 126] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(228, 4)[4]   # 228.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 228:4 1000