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

Newform orbit 38.6.a.c

Trace form

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

$2q - 8q^{2} + 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} - 128q^{8} + 239q^{9} + 180q^{10} + 661q^{11} + 48q^{12} + 1613q^{13} - 456q^{14} + 2094q^{15} + 512q^{16} + 64q^{17} - 956q^{18} + 722q^{19} - 720q^{20} - 2711q^{21} - 2644q^{22} - 3185q^{23} - 192q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1441})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 42x - 5323$

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
$\operatorname{tr} a_p$	−8	3	−45	114	661	1613	64	722	−3185	−2481	−1180	10488	16630	11303	−12155	20585	−78581	43621	7805	−62488	16218	67122	−10714	128188	178558

$\operatorname{tr} a_n$ 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)[2]   # 38.6.a.c

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

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