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

Newform orbit 871.2.a.f

Trace form

The 23 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$23q + 4q^{2} + 4q^{3} + 26q^{4} + 6q^{5} - 3q^{6} - 2q^{7} + 12q^{8} + 49q^{9} - 7q^{10} + 7q^{11} + 7q^{12} + 23q^{13} + 9q^{14} + 2q^{15} + 24q^{16} + 28q^{17} - 4q^{18} - q^{19} + 4q^{20} + 7q^{21} - 6q^{22} + 20q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 23. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{23} - 8x^{22} - 60x^{21} + 610x^{20} + 919x^{19} - 17331x^{18} + 6158x^{17} + 235093x^{16} - 267843x^{15} - 1699634x^{14} + 2502496x^{13} + 7118905x^{12} - 10914842x^{11} - 18487636x^{10} + 25407834x^{9} + 30528661x^{8} - 32465613x^{7} - 31246998x^{6} + 22332731x^{5} + 18672541x^{4} - 7667302x^{3} - 5854232x^{2} + 1021734x + 733790$

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
$\operatorname{tr} a_p$	4	4	6	−2	7	23	28	−1	20	55	−11	10	28	−3	9	47	17	−1	−23	42	5	−6	−18	68	6

$\operatorname{tr} a_n$ 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)[5]   # 871.2.a.f

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

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