Atlas › Modular forms › Weight 12 › Level 19 › 19.12.a › 19.12.a.b

Newform orbit 19.12.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 87q^{2} + 496q^{3} + 11781q^{4} + 2114q^{5} + 84181q^{6} - 19080q^{7} - 76701q^{8} + 632433q^{9} - 268428q^{10} + 175744q^{11} + 3993305q^{12} + 809154q^{13} + 7041403q^{14} + 4320440q^{15} + 29590233q^{16} + 13162874q^{17} + 51670980q^{18} - 22284891q^{19} + 692380q^{20} + 75424388q^{21} + 63849138q^{22} + 114786388q^{23} + 310823511q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 583x^{8} - 1042721x^{7} + 466702653x^{6} + 396196150308x^{5} - 103292325596454x^{4} - 59342910232666924x^{3} + 3714367199125112392x^{2} + 1458964133140235097008x - 50351612334872711003904$

Atkin–Lehner signs

$p$	$w_p$
19	$+1$

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
$\operatorname{tr} a_p$	87	496	2114	−19080	175744	809154	13162874	−22284891	114786388	54612914	−226600416	−371753970	504959690	−1620951792	−1965882540	14154895258	22029386080	−1263984246	14718513264	34967811888	18297073746	−15944238816	−2858023276	207569871794	58127902530

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.506 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, 19]: [2, 20] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 6 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(19, 12)[1]   # 19.12.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 19:12 1000