Atlas › Modular forms › Weight 2 › Level 871 › 871.2.a › 871.2.a.c

Newform orbit 871.2.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + 8x^{7} + 10x^{6} - 56x^{5} - 133x^{4} + 9x^{3} + 142x^{2} + 69x + 9$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
67	$-1$

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
$\operatorname{tr} a_p$	−4	−4	−8	0	−3	8	−16	−5	−8	−13	1	6	−18	−17	9	−33	7	1	8	−40	3	2	−8	−20	−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 0.248 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, 13, 67, 871]: [0, 1, 6, 80] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (phi(ord eps) x dim S_k^new); 7 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(871, 2)[2]   # 871.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 871:2 1000