Atlas › Modular forms › Weight 2 › Level 886 › 886.2.a › 886.2.a.k

Newform orbit 886.2.a.k

Trace form

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

$9q - 9q^{2} - 4q^{3} + 9q^{4} + 9q^{5} + 4q^{6} - 6q^{7} - 9q^{8} + 17q^{9} - 9q^{10} + 14q^{11} - 4q^{12} + 7q^{13} + 6q^{14} - 10q^{15} + 9q^{16} + 4q^{17} - 17q^{18} - 4q^{19} + 9q^{20} + 9q^{21} - 14q^{22} + 6q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

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

$x^{9} - 23x^{8} + 43x^{7} + 2252x^{6} - 13783x^{5} - 35541x^{4} + 413903x^{3} - 422113x^{2} - 2295014x + 4139063$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
443	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	−9	−4	9	−6	14	7	4	−4	6	19	16	5	20	−5	11	18	35	25	−21	40	−9	5	23	54	−18

$\operatorname{tr} a_n$ 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, 443, 886]: [0, 1, 38, 113] (certified at ell = 2147481311)
T = [(3, 1), (5, 3)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 6 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(886, 2)[10]   # 886.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 886:2 1000