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

Newform orbit 38.8.a.c

Trace form

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

$2q - 16q^{2} - 11q^{3} + 128q^{4} - 69q^{5} + 88q^{6} - 348q^{7} - 1024q^{8} + 4663q^{9} + 552q^{10} + 2723q^{11} - 704q^{12} - 14113q^{13} + 2784q^{14} - 26550q^{15} + 8192q^{16} - 38560q^{17} - 37304q^{18} + 13718q^{19} - 4416q^{20} - 123757q^{21} - 21784q^{22} - 73897q^{23} + 5632q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17953})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 80x - 16353$

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 odd 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	−11	−69	−348	2723	−14113	−38560	13718	−73897	159813	−259468	−528168	1005650	286217	−1397509	−1385969	2700953	−3975947	−134557	4202740	900498	−6893730	−2465330	17431724	6351934

$\operatorname{tr} a_n$ 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)[2]   # 38.8.a.c

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

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