Atlas › Modular forms › Weight 4 › Level 56 › 56.4.a

Newspace 56.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(56))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
56.4.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 16q^{5} - 7q^{7} + O(q^{8})$	+
56.4.a.b	1	$\mathbb{Q}$	$q + 6q^{3} + 8q^{5} - 7q^{7} + O(q^{8})$	+
56.4.a.c	2	$\mathbb{Q}(\sqrt{57})$	$2q - 2q^{3} + 22q^{5} + 14q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
56.4.a.a	$x + 50$
56.4.a.b	$x - 30$
56.4.a.c	$x^{2} - 64x + 796$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(56))$	4
Old forms	16
Cusp forms $S_{4}(\Gamma_0(56))$	20
Eisenstein series $E_{4}(\Gamma_0(56))$	8
Modular forms $M_{4}(\Gamma_0(56))$	28

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(56)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(28))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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, 4, 7, 8, 14, 28, 56]: [1, 2, 3, 3, 5, 8, 15, 28] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(56, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 56:4 1000

← 55.4.a · 57.4.a →