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

Newform orbit 232.4.a.d

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 4q^{3} + 10q^{5} + 32q^{7} + 29q^{9} + 36q^{11} + 26q^{13} + 88q^{15} + 82q^{17} + 156q^{19} + 72q^{21} + 336q^{23} + O(q^{25})$

Coefficient field

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

$x^{5} - 14x^{4} - 460x^{3} + 5600x^{2} - 732x - 2664$

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	4	10	32	36	26	82	156	336	145	432	−18	82	340	680	−102	924	−618	44	1032	−1078	200	452	−1790	−2518

$\operatorname{tr} a_n$ 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)[3]   # 232.4.a.d

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

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