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

Newform orbit 19.10.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 15q^{2} + 7q^{3} + 2645q^{4} + 3894q^{5} - 9723q^{6} - 7133q^{7} + 10911q^{8} + 102715q^{9} + 113172q^{10} + 172818q^{11} + 349117q^{12} + 109291q^{13} + 250959q^{14} + 457332q^{15} + 1590377q^{16} + 583575q^{17} + 308688q^{18} + 1042568q^{19} + 3191676q^{20} + 372767q^{21} + 57234q^{22} - 2292405q^{23} - 7378809q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 22x^{7} - 123495x^{6} + 6750378x^{5} + 4471450734x^{4} - 329333903580x^{3} - 37184855974632x^{2} + 1827374957035920x + 64654738149362688$

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	7	3894	−7133	172818	109291	583575	1042568	−2292405	4546335	−8277416	24485536	35051946	36624268	9865524	13963683	21215463	180552112	1767355	273631578	−263711543	744472174	832139910	−291457734	−3483837140

$\operatorname{tr} a_n$ 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)[1]   # 19.10.a.b

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

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