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

Newform orbit 47.6.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 9q^{2} - 20q^{3} + 53q^{4} - 58q^{5} - 32q^{6} - 236q^{7} - 327q^{8} + 129q^{9} - 600q^{10} - 436q^{11} - 2340q^{12} - 2304q^{13} - 2326q^{14} - 2554q^{15} - 1871q^{16} - 2244q^{17} - 2663q^{18} - 882q^{19} + 688q^{20} - 2412q^{21} - 2086q^{22} - 1842q^{23} + 7416q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 29x^{6} - 601x^{5} - 13429x^{4} + 83355x^{3} + 1390049x^{2} - 3319032x - 10267716$

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
$\operatorname{tr} a_p$	−9	−20	−58	−236	−436	−2304	−2244	−882	−1842	5130	2388	−13944	−10820	−24584	−15463	−55104	−58828	−28992	−90932	61508	−107256	−63144	108956	51832	−153476

$\operatorname{tr} a_n$ 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)[0]   # 47.6.a.a

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

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