Atlas › Modular forms › Weight 2 › Level 887 › 887.2.a › 887.2.a.a

Newform orbit 887.2.a.a

Trace form

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

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

Coefficient field

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

$x^{2} - 3x + 1$

Atkin–Lehner signs

$p$	$w_p$
887	$+1$

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
$\operatorname{tr} a_p$	1	2	−3	−2	−3	3	5	−2	0	−9	−14	1	−14	11	17	6	−8	−9	−2	−14	−1	6	3	6	10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 887]: [0, 75] (certified at ell = 2147476169)
T = [(2, 1), (3, 1)]; h of degree 74 = 1 x 74 (phi(ord eps) x dim S_k^new); 8 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(887, 2)[0]   # 887.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 887:2 1000