Atlas › Modular forms › Weight 6 › Level 47 › 47.6.a › 47.6.a.b

Newform orbit 47.6.a.b

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 7q^{2} + 16q^{3} + 213q^{4} + 42q^{5} + 112q^{6} + 352q^{7} + 441q^{8} + 1344q^{9} + 400q^{10} + 290q^{11} + 2693q^{12} + 2166q^{13} + 4923q^{14} + 3336q^{15} + 6593q^{16} + 3188q^{17} + 3642q^{18} + 3518q^{19} - 2064q^{20} - 1576q^{21} + 1200q^{22} - 2660q^{23} - 6637q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 23x^{11} - 2276x^{10} + 45314x^{9} + 1866389x^{8} - 31980858x^{7} - 676169549x^{6} + 9716033175x^{5} + 101210907638x^{4} - 1122526657354x^{3} - 3769953988577x^{2} + 13345867482952x - 8726123100764$

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 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
$\operatorname{tr} a_p$	7	16	42	352	290	2166	3188	3518	−2660	6910	−492	27704	−37292	14222	26508	39120	18068	33496	94522	63920	188880	−62076	−90524	−6228	92828

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.226 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, 21] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 4 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, 6)[1]   # 47.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 47:6 1000