Atlas › Modular forms › Weight 6 › Level 19 › 19.6.a › 19.6.a.c

Newform orbit 19.6.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 7q^{2} - 7q^{3} + 49q^{4} - 133q^{5} - 241q^{6} + 72q^{7} - 567q^{8} + 335q^{9} + 908q^{10} - 705q^{11} + 1687q^{12} - 1341q^{13} + 987q^{14} - 862q^{15} + 1921q^{16} + 2784q^{17} + 686q^{18} - 722q^{19} - 6356q^{20} - 3969q^{21} + 3618q^{22} - 2713q^{23} - 2529q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{177})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 14x - 128$

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
$\operatorname{tr} a_p$	−7	−7	−133	72	−705	−1341	2784	−722	−2713	−7775	7132	−6248	−4174	25357	11727	−29133	−64515	−40939	19039	−70236	67058	−32850	71534	−87268	−62458

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.211 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, 10] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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, 6)[2]   # 19.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 19:6 1000