Atlas › Modular forms › Weight 6 › Level 38 › 38.6.a

Newspace 38.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(38))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
38.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 6q^{3} + 16q^{4} + 31q^{5} + 24q^{6} - 27q^{7} + O(q^{8})$	+ +	$-1$
38.6.a.b	1	$\mathbb{Q}$	$q + 4q^{2} - 14q^{3} + 16q^{4} - 45q^{5} - 56q^{6} - 121q^{7} + O(q^{8})$	− −	$-1$
38.6.a.c	2	$\mathbb{Q}(\sqrt{1441})$	$2q - 8q^{2} + 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} + O(q^{8})$	+ −	$+1$
38.6.a.d	3	degree 3	$3q + 12q^{2} + 13q^{3} + 48q^{4} + 81q^{5} + 52q^{6} + 228q^{7} + O(q^{8})$	− +	$+1$

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 $T$
38.6.a.a	$x - 25$
38.6.a.b	$x + 59$
38.6.a.c	$x^{2} + 42x - 5323$
38.6.a.d	$x^{3} - 94x^{2} + 1859x + 9802$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(38))$	7
Old forms	16
Cusp forms $S_{6}(\Gamma_0(38))$	23
Eisenstein series $E_{6}(\Gamma_0(38))$	4
Modular forms $M_{6}(\Gamma_0(38))$	27

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(38)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 10, 27] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 38:6 1000

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