Atlas › Modular forms › Weight 2 › Level 885 › 885.2.a › 885.2.a.k

Newform orbit 885.2.a.k

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7}$ , whose minimal polynomial is

$x^{6} - 20x^{5} - 22x^{4} + 1830x^{3} - 1131x^{2} - 44890x - 16706$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$+1$
59	$+1$

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
$\operatorname{tr} a_p$	2	6	−6	6	12	16	2	8	−8	−8	8	8	0	16	8	−2	−6	−4	4	−10	24	−32	0	−8	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.438 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, 5, 15, 59, 177, 295, 885]: [0, 1, 1, 4, 6, 22, 32, 124] (certified at ell = 2147482481)
T = [(2, 1), (7, 3)]; h of degree 39 = 1 x 39 (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(885, 2)[10]   # 885.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 885:2 1000