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

Newform orbit 871.2.a.e

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 5q^{2} + 23q^{4} + 10q^{5} + 3q^{6} + 21q^{8} + 38q^{9} - q^{10} + 19q^{11} - 7q^{12} - 20q^{13} + 13q^{14} + 4q^{15} + 41q^{16} + 11q^{18} + 9q^{19} + 20q^{20} + 5q^{21} + 6q^{22} - 2q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{20} - 5x^{19} - 71x^{18} + 361x^{17} + 1900x^{16} - 9839x^{15} - 24739x^{14} + 130056x^{13} + 170877x^{12} - 900727x^{11} - 626997x^{10} + 3308844x^{9} + 1068970x^{8} - 6350364x^{7} - 527544x^{6} + 5928735x^{5} - 223248x^{4} - 2346028x^{3} + 144695x^{2} + 233658x + 22834$

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 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
$\operatorname{tr} a_p$	5	0	10	0	19	−20	0	9	−2	11	23	−16	66	−11	3	7	47	3	20	42	1	6	−14	60	−4

$\operatorname{tr} a_n$ 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)[4]   # 871.2.a.e

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

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