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

Newform orbit 19.8.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 9q^{2} - 14q^{3} + 37q^{4} - 222q^{5} - 603q^{6} - 1246q^{7} - 3555q^{8} - 4898q^{9} - 6444q^{10} - 8718q^{11} - 6281q^{12} - 4480q^{13} - 1935q^{14} - 2760q^{15} + 19393q^{16} - 4440q^{17} + 52722q^{18} + 27436q^{19} + 81228q^{20} + 34124q^{21} + 115182q^{22} - 30528q^{23} + 90135q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 23x^{3} - 1332x^{2} - 12312x + 415872$

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 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	−14	−222	−1246	−8718	−4480	−4440	27436	−30528	−254244	−303460	−270460	−828564	37454	335670	76728	−3191334	346550	−270322	−2066124	−416044	16025864	8524128	2899092	−4766908

$\operatorname{tr} a_n$ 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)[0]   # 19.8.a.a

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

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