Atlas › Modular forms › Weight 6 › Level 38 › 38.6.a › 38.6.a.d

Newform orbit 38.6.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 12q^{2} + 13q^{3} + 48q^{4} + 81q^{5} + 52q^{6} + 228q^{7} + 192q^{8} + 236q^{9} + 324q^{10} + 363q^{11} + 208q^{12} + 501q^{13} + 912q^{14} - 670q^{15} + 768q^{16} - 1206q^{17} + 944q^{18} - 1083q^{19} + 1296q^{20} - 2085q^{21} + 1452q^{22} - 1077q^{23} + 832q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{3} - 94x^{2} + 1859x + 9802$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
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
$\operatorname{tr} a_p$	12	13	81	228	363	501	−1206	−1083	−1077	−8349	−7332	−1650	10140	3777	33231	31029	20409	17115	−789	19164	−76260	68358	6762	−85506	105024

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 2, 19, 38]: [1, 2, 10, 27] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(38, 6)[3]   # 38.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 38:6 1000