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

Newform orbit 19.6.a.d

Trace form

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

$4q + 9q^{2} + 6q^{3} + 97q^{4} + 90q^{5} + 369q^{6} - 190q^{7} - 225q^{8} + 846q^{9} - 264q^{10} - 162q^{11} - 2091q^{12} - 52q^{13} + 1395q^{14} - 1944q^{15} - 551q^{16} - 288q^{17} - 7506q^{18} + 1444q^{19} + 900q^{20} + 6096q^{21} - 222q^{22} + 9900q^{23} + 3879q^{24} + O(q^{25})$

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

$x^{4} - 15x^{3} - 1278x^{2} + 25974x - 82176$

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
$\operatorname{tr} a_p$	9	6	90	−190	−162	−52	−288	1444	9900	4176	13580	1172	9540	13370	28098	34740	9702	−37978	−2974	32220	−86908	−165736	−146448	−26604	313820

$\operatorname{tr} a_n$ 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)[3]   # 19.6.a.d

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

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