Atlas › Modular forms › Weight 2 › Level 847 › 847.2.a › 847.2.a.e

Newform orbit 847.2.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - q^{2} - q^{3} + 3q^{4} + 7q^{6} + 2q^{7} - 6q^{8} + q^{9} - 13q^{10} - 8q^{12} - 6q^{13} - q^{14} - 13q^{15} - 3q^{16} - 9q^{17} - 7q^{18} + 6q^{19} + 13q^{20} - q^{21} - 9q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{13}$ , whose minimal polynomial is

$x^{2} + 46x + 321$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
11	— ( $11^2 \mid 847$ )

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
$\operatorname{tr} a_p$	−1	−1	0	2	0	−6	−9	6	−9	−13	2	4	−14	−7	7	−15	17	5	−1	−5	10	13	−6	3	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.343 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, 7, 11, 77, 121, 847]: [0, 1, 2, 10, 12, 90] (certified at ell = 2147482921)
T = [(2, 1), (3, 3), (5, 5), (13, 7)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 12 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(847, 2)[4]   # 847.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 847:2 1000