Atlas › Modular forms › Weight 8 › Level 19 › 19.8.a › 19.8.a.b

Newform orbit 19.8.a.b

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 15q^{2} + 40q^{3} + 357q^{4} + 219q^{5} + 925q^{6} + 2105q^{7} + 5835q^{8} + 5916q^{9} + 8212q^{10} + 7257q^{11} + 7025q^{12} + 6850q^{13} + 8859q^{14} + 3650q^{15} - 9159q^{16} + 5415q^{17} - 58980q^{18} - 41154q^{19} - 67620q^{20} - 83290q^{21} - 223870q^{22} - 720q^{23} - 151113q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 55x^{5} - 9494x^{4} + 463630x^{3} + 23057000x^{2} - 985214600x - 3813684864$

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
$\operatorname{tr} a_p$	15	40	219	2105	7257	6850	5415	−41154	−720	381624	264080	1082300	485232	198705	−247125	3226770	2305380	585731	−3264030	6833682	−4160625	−8680576	−3785040	12473466	882830

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.265 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]: [1, 12] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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, 8)[1]   # 19.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 19:8 1000