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

Newform orbit 38.10.a.d

Trace form

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

$4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + 22320q^{10} - 104249q^{11} + 21504q^{12} + 120486q^{13} - 196912q^{14} - 591090q^{15} + 262144q^{16} - 412139q^{17} - 264608q^{18} + 521284q^{19} - 357120q^{20} + 2437006q^{21} + 1667984q^{22} + 3010300q^{23} - 344064q^{24} + O(q^{25})$

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

$x^{4} + 1311x^{3} - 7383727x^{2} - 6016162083x + 12629431516338$

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	84	−1395	12307	−104249	120486	−412139	521284	3010300	6153240	12774024	20506048	11620300	7698327	−31581083	72549422	−149234120	129004373	132595266	−47138482	−39332795	−307010840	−746568232	286943482	793519958

$\operatorname{tr} a_n$ 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)[3]   # 38.10.a.d

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

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