Atlas › Modular forms › Weight 4 › Level 47 › 47.4.a › 47.4.a.b

Newform orbit 47.4.a.b

Trace form

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

$8q + 3q^{2} + 7q^{3} + 45q^{4} + 14q^{5} + 16q^{6} + 39q^{7} + 57q^{8} + 93q^{9} + 60q^{10} + 68q^{11} - 19q^{12} + 112q^{13} - 277q^{14} - 126q^{15} - 87q^{16} - 7q^{17} - 474q^{18} + 74q^{19} - 432q^{20} - 160q^{21} + 20q^{22} + 64q^{23} - 325q^{24} + O(q^{25})$

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

$x^{8} - 10x^{7} - 175x^{6} + 2193x^{5} + 3021x^{4} - 111350x^{3} + 383800x^{2} - 203499x - 228937$

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
$\operatorname{tr} a_p$	3	7	14	39	68	112	−7	74	64	20	176	811	466	1164	−376	747	−1661	1051	174	−631	1982	−2387	−3064	1167	1901

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.093 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, 13] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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, 4)[1]   # 47.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 47:4 1000