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

Newform orbit 47.8.a.b

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 15q^{2} + 40q^{3} + 1125q^{4} + 444q^{5} - 128q^{6} + 1860q^{7} + 2145q^{8} + 16986q^{9} + 7680q^{10} + 4776q^{11} - 20035q^{12} + 16074q^{13} - 7749q^{14} + 22614q^{15} + 131873q^{16} + 67106q^{17} + 243090q^{18} + 69730q^{19} + 300392q^{20} + 247700q^{21} + 357976q^{22} + 101110q^{23} + 167915q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 55x^{15} - 25935x^{14} + 1076227x^{13} + 282916538x^{12} - 7798781507x^{11} - 1646865483912x^{10} + 24377050010755x^{9} + 5364546862289334x^{8} - 24374516395924541x^{7} - 9488679642127527088x^{6} - 22232369033877521159x^{5} + 8443102437779313561741x^{4} + 48715246409229699937466x^{3} - 3256621975957581031146953x^{2} - 15014623430239403913789432x + 473419371800796159567608964$

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
$\operatorname{tr} a_p$	15	40	444	1860	4776	16074	67106	69730	101110	199976	373036	596106	−66678	1037964	−1661168	−1253454	−1977256	−2381646	11455752	5467788	12652462	34366848	27856008	14703638	51072842

$\operatorname{tr} a_n$ 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)[1]   # 47.8.a.b

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

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