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

Newform orbit 886.2.a.i

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{6} + 18x^{5} + 17x^{4} - 713x^{3} - 879x^{2} + 464x + 311$

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 odd 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
$\operatorname{tr} a_p$	−6	3	−7	0	−12	−8	−12	−10	2	−2	−2	−10	−37	18	−14	−6	−36	14	10	−23	29	10	−22	−25	−26

$\operatorname{tr} a_n$ 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)[8]   # 886.2.a.i

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

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