Atlas › Modular forms › Weight 2 › Level 891 › 891.2.a › 891.2.a.j

Newform orbit 891.2.a.j

Trace form

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

$2q + q^{2} + 3q^{4} + q^{5} + 6q^{8} + 7q^{10} + 2q^{11} - 9q^{13} + 13q^{14} - 3q^{16} - 2q^{17} - 7q^{19} + 8q^{20} + q^{22} - 4q^{23} + O(q^{25})$

Coefficient field

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

$x^{2} + 59x + 711$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 891$ )
11	$-1$

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
$\operatorname{tr} a_p$	1	0	1	0	2	−9	−2	−7	−4	3	15	8	13	−16	−1	9	20	8	13	3	−6	1	12	−3	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.406 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, 3, 9, 11, 27, 33, 81, 99, 297, 891]: [0, 1, 2, 2, 4, 6, 10, 14, 38, 110] (certified at ell = 2147482801)
T = [(2, 1), (5, 3), (7, 5), (13, 7)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 9 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(891, 2)[9]   # 891.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 891:2 1000