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

Newspace 38.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(38))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
38.10.a.a	1	$\mathbb{Q}$	$q + 16q^{2} - 119q^{3} + 256q^{4} - 684q^{5} - 1904q^{6} + 9149q^{7} + O(q^{8})$	− −	$-1$
38.10.a.b	1	$\mathbb{Q}$	$q + 16q^{2} + 102q^{3} + 256q^{4} - 1581q^{5} + 1632q^{6} - 4865q^{7} + O(q^{8})$	− −	$-1$
38.10.a.c	3	degree 3	$3q - 48q^{2} + 3q^{3} + 768q^{4} + 486q^{5} - 48q^{6} - 13317q^{7} + O(q^{8})$	+ +	$-1$
38.10.a.d	4	degree 4	$4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} + O(q^{8})$	+ −	$+1$
38.10.a.e	4	degree 4	$4q + 64q^{2} + 226q^{3} + 1024q^{4} + 866q^{5} + 3616q^{6} + 2670q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
38.10.a.a	$x + 803$
38.10.a.b	$x + 1479$
38.10.a.c	$x^{3} - 489x^{2} - 584001x - 100504663$
38.10.a.d	$x^{4} + 1311x^{3} - 7383727x^{2} - 6016162083x + 12629431516338$
38.10.a.e	$x^{4} - 1092x^{3} - 3864974x^{2} + 1096594884x + 2934994889485$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(38))$	13
Old forms	30
Cusp forms $S_{10}(\Gamma_0(38))$	43
Eisenstein series $E_{10}(\Gamma_0(38))$	4
Modular forms $M_{10}(\Gamma_0(38))$	47

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(38)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

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)

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

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

← 37.10.a · 39.10.a →