Atlas › Modular forms › Weight 4 › Level 232 › 232.4.a › 232.4.a.a

Newform orbit 232.4.a.a

Trace form

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

$3q + 3q^{3} + 11q^{5} - 38q^{7} + 58q^{9} + 65q^{11} + 29q^{13} - 85q^{15} + 244q^{17} + 312q^{19} - 214q^{21} - 24q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} - 14x^{2} - 292x + 3576$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 232$ )
29	$+1$

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	11	−38	65	29	244	312	−24	−87	249	200	−470	97	377	1007	−364	−524	−820	−782	1620	427	1520	−2474	−170

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.153 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, 29, 58, 116, 232]: [1, 2, 3, 5, 9, 25, 48, 94] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 4 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(232, 4)[0]   # 232.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 232:4 1000