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

Newform orbit 38.8.a.d

Trace form

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

$2q + 16q^{2} - 69q^{3} + 128q^{4} + 155q^{5} - 552q^{6} - 2238q^{7} + 1024q^{8} + 855q^{9} + 1240q^{10} - 3295q^{11} - 4416q^{12} - 13427q^{13} - 17904q^{14} - 34782q^{15} + 8192q^{16} - 32256q^{17} + 6840q^{18} - 13718q^{19} + 9920q^{20} + 103797q^{21} - 26360q^{22} - 82525q^{23} - 35328q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{633})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 86x - 122219$

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	−69	155	−2238	−3295	−13427	−32256	−13718	−82525	−12749	258944	−149260	339130	−83869	1471025	−945643	−969009	−1506755	−1848219	−3417184	−2499822	2636926	−10059354	−3506160	5893526

$\operatorname{tr} a_n$ 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)[3]   # 38.8.a.d

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

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