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

Newform orbit 847.2.a.p

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + q^{2} + 4q^{3} + 7q^{4} + 10q^{5} - q^{6} + 8q^{7} + 14q^{9} + 6q^{10} + 9q^{12} - 6q^{13} + q^{14} + 11q^{15} + q^{16} - 5q^{17} + 8q^{18} - 13q^{19} + 23q^{20} + 4q^{21} + 16q^{23} - 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{13}$ , whose minimal polynomial is

$x^{8} - 21x^{7} - 3591x^{6} + 63670x^{5} + 3142135x^{4} - 27778580x^{3} - 805511250x^{2} - 517825450x + 10712525275$

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	4	10	8	0	−6	−5	−13	16	9	9	7	−10	−4	16	37	1	19	−19	13	−25	0	−25	37	15

$\operatorname{tr} a_n$ 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)[15]   # 847.2.a.p

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

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