Atlas › Modular forms › Weight 10 › Level 38 › 38.10.a › 38.10.a.e

Newform orbit 38.10.a.e

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 64q^{2} + 226q^{3} + 1024q^{4} + 866q^{5} + 3616q^{6} + 2670q^{7} + 16384q^{8} + 50606q^{9} + 13856q^{10} + 119234q^{11} + 57856q^{12} - 6748q^{13} + 42720q^{14} + 355904q^{15} + 262144q^{16} + 678624q^{17} + 809696q^{18} - 521284q^{19} + 221696q^{20} + 1586736q^{21} + 1907744q^{22} + 2911868q^{23} + 925696q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} - 1092x^{3} - 3864974x^{2} + 1096594884x + 2934994889485$

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
$\operatorname{tr} a_p$	64	226	866	2670	119234	−6748	678624	−521284	2911868	8291104	3445468	−1005524	8514124	13900726	−36334954	−113969356	−396773766	−298192066	−113551722	4659620	136198452	67255424	1376505216	1557211260	975818188

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.333 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, 16, 47] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 5 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, 10)[4]   # 38.10.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 38:10 1000