Atlas › Modular forms › Weight 4 › Level 38 › 38.4.a › 38.4.a.c

Newform orbit 38.4.a.c

Trace form

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

$2q + 4q^{2} + 9q^{3} + 8q^{4} - 9q^{5} + 18q^{6} - 18q^{7} + 16q^{8} + 23q^{9} - 18q^{10} - 17q^{11} + 36q^{12} + 17q^{13} - 36q^{14} - 150q^{15} + 32q^{16} - 80q^{17} + 46q^{18} + 38q^{19} - 36q^{20} - 227q^{21} - 34q^{22} + 73q^{23} + 72q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{73})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 73$

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
$\operatorname{tr} a_p$	4	9	−9	−18	−17	17	−80	38	73	3	212	192	−50	677	−389	−1219	−287	313	1223	200	378	1350	−670	−236	1294

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.1 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, 6, 17] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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, 4)[2]   # 38.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 38:4 1000