Atlas › Modular forms › Weight 8 › Level 38 › 38.8.a › 38.8.a.b

Newform orbit 38.8.a.b

Trace form

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

$2q - 16q^{2} - 61q^{3} + 128q^{4} + 175q^{5} + 488q^{6} - 2592q^{7} - 1024q^{8} - 1145q^{9} - 1400q^{10} + 1045q^{11} - 3904q^{12} + 14647q^{13} + 20736q^{14} + 15190q^{15} + 8192q^{16} + 29616q^{17} + 9160q^{18} - 13718q^{19} + 11200q^{20} + 87267q^{21} - 8360q^{22} + 69985q^{23} + 31232q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2737})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 114x - 171919$

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
$\operatorname{tr} a_p$	−16	−61	175	−2592	1045	14647	29616	−13718	69985	138821	−199396	−67840	−539350	602639	−1031975	2138263	−3936369	−1027655	764949	−3572084	9069522	−2753414	−7643046	1393620	−6921466

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 3, 12, 37] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (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(38, 8)[1]   # 38.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 38:8 1000