Atlas › Modular forms › Weight 2 › Level 932 › 932.2.a › 932.2.a.b

Newform orbit 932.2.a.b

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 6q^{3} + 5q^{5} - q^{7} + 21q^{9} + 3q^{11} + 20q^{13} + 2q^{15} + 15q^{17} + 10q^{19} + 10q^{21} + 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{13} - 11x^{12} - 19x^{11} + 537x^{10} - 685x^{9} - 8776x^{8} + 20302x^{7} + 55788x^{6} - 174450x^{5} - 88071x^{4} + 508501x^{3} - 191557x^{2} - 165902x - 17542$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 932$ )
233	$+1$

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	6	5	−1	3	20	15	10	4	11	2	34	16	7	4	17	9	18	26	−11	21	−26	−4	1	53

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 233, 466, 932]: [0, 1, 2, 20, 60, 120] (certified at ell = 2147482481)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (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(932, 2)[1]   # 932.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 932:2 1000