Atlas › Modular forms › Weight 8 › Level 47 › 47.8.a › 47.8.a.a

Newform orbit 47.8.a.a

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q - 17q^{2} - 68q^{3} + 485q^{4} - 56q^{5} - 992q^{6} - 2256q^{7} - 3999q^{8} + 6051q^{9} - 2320q^{10} - 3210q^{11} + 10044q^{12} - 16468q^{13} + 7090q^{14} - 35656q^{15} - 26863q^{16} - 70358q^{17} - 204335q^{18} - 92830q^{19} - 172472q^{20} - 160896q^{21} - 116166q^{22} - 128236q^{23} - 440688q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 85x^{10} - 11396x^{9} - 776664x^{8} + 49741911x^{7} + 2199688997x^{6} - 94915166122x^{5} - 1946034505916x^{4} + 64033249957883x^{3} + 292047342823529x^{2} - 3552085556855880x - 886277305483956$

Atkin–Lehner signs

$p$	$w_p$
47	$-1$

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
$\operatorname{tr} a_p$	−17	−68	−56	−2256	−3210	−16468	−70358	−92830	−128236	−122364	47212	−1185510	375714	−815686	1142053	146178	2234648	−3090662	−4567986	−3407112	−13224794	−18518452	−19632736	−16757438	−34129206

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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 engine's checks

dim M^+ at levels [1, 47]: [1, 29] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (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(47, 8)[0]   # 47.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 47:8 1000