Atlas › Modular forms › Weight 2 › Level 824 › 824.2.a › 824.2.a.e

Newform orbit 824.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 7q^{5} + q^{7} + 13q^{9} + q^{11} + 14q^{13} + 3q^{15} + 2q^{17} + 6q^{19} + 18q^{21} - 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{9} - 159x^{8} - 3422x^{7} + 1428807x^{6} - 30314200x^{5} - 3625686447x^{4} + 139937303251x^{3} + 1677757391290x^{2} - 118330251693084x + 966600405539240$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 824$ )
103	$+1$

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	0	7	1	1	14	2	6	−9	29	12	18	5	4	3	41	2	14	8	5	−7	4	−32	8	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.394 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, 8, 103, 206, 412, 824]: [0, 1, 2, 3, 9, 28, 55, 108] (certified at ell = 2147482501)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 7 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(824, 2)[4]   # 824.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 824:2 1000