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

Newform orbit 38.8.a.e

Trace form

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

$4q + 32q^{2} + 12q^{3} + 256q^{4} - 279q^{5} + 96q^{6} + 2485q^{7} + 2048q^{8} + 9482q^{9} - 2232q^{10} + 5269q^{11} + 768q^{12} + 5406q^{13} + 19880q^{14} + 26658q^{15} + 16384q^{16} + 22885q^{17} + 75856q^{18} + 27436q^{19} - 17856q^{20} + 2854q^{21} + 42152q^{22} + 3364q^{23} + 6144q^{24} + O(q^{25})$

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

$x^{4} + 267x^{3} - 212659x^{2} - 31505691x + 5724320994$

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
$\operatorname{tr} a_p$	32	12	−279	2485	5269	5406	22885	27436	3364	−122136	225480	154096	−1054628	−840795	−1021877	−326842	421384	116825	5794566	10590626	3971389	5597800	4665800	−2794214	−14445130

$\operatorname{tr} a_n$ 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)[4]   # 38.8.a.e

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

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