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

Newform orbit 47.4.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 5q^{2} - 5q^{3} + 5q^{4} - 6q^{5} - 8q^{6} - 45q^{7} - 39q^{8} - 42q^{9} - 40q^{10} + 2q^{11} + 12q^{12} - 80q^{13} + 162q^{14} + 14q^{15} + 89q^{16} - 39q^{17} + 181q^{18} - 24q^{19} + 232q^{20} + 24q^{21} - 14q^{22} + 120q^{23} + 192q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 10x^{2} + 24x - 3$

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
$\operatorname{tr} a_p$	−5	−5	−6	−45	2	−80	−39	−24	120	−184	−4	−589	−92	−250	141	459	579	267	−540	749	−1924	805	712	835	−2243

$\operatorname{tr} a_n$ 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)[0]   # 47.4.a.a

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

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