Atlas › Modular forms › Weight 10 › Level 19 › 19.10.a › 19.10.a.a

Newform orbit 19.10.a.a

Trace form

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

$6q - 33q^{2} - 155q^{3} + 1365q^{4} - 3612q^{5} + 5581q^{6} + 4085q^{7} - 23511q^{8} + 17625q^{9} - 93884q^{10} - 69312q^{11} - 460165q^{12} - 191747q^{13} - 644691q^{14} - 326428q^{15} + 13905q^{16} - 288195q^{17} - 1159386q^{18} - 781926q^{19} - 1551444q^{20} - 3260767q^{21} + 2409710q^{22} - 50697q^{23} + 5457255q^{24} + O(q^{25})$

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

$x^{6} + 188x^{5} - 57989x^{4} - 10000148x^{3} + 459954260x^{2} + 92468760928x + 1678674266112$

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
$\operatorname{tr} a_p$	−33	−155	−3612	4085	−69312	−191747	−288195	−781926	−50697	−7178667	−4310716	−15687040	−25134306	−75118674	−83731938	−138019203	−7809915	191946566	−16109787	−264469698	−287572857	−86534002	−359214570	−2263866306	2705893460

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.294 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, 16] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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, 10)[0]   # 19.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 19:10 1000