Atlas › Modular forms › Weight 4 › Level 228 › 228.4.a

Newspace 228.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(228))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
228.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 7q^{5} + 21q^{7} + O(q^{8})$	+ +
228.4.a.b	1	$\mathbb{Q}$	$q - 3q^{3} + 4q^{5} - 12q^{7} + O(q^{8})$	+ +
228.4.a.c	1	$\mathbb{Q}$	$q + 3q^{3} - 3q^{5} - 17q^{7} + O(q^{8})$	− −
228.4.a.d	2	$\mathbb{Q}(\sqrt{897})$	$2q - 6q^{3} + 3q^{5} - 17q^{7} + O(q^{8})$	+ −
228.4.a.e	3	degree 3	$3q + 9q^{3} + 31q^{5} + 9q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
228.4.a.a	$x - 14$
228.4.a.b	$x + 8$
228.4.a.c	$x + 20$
228.4.a.d	$x^{2} + 14x - 848$
228.4.a.e	$x^{3} - 40x^{2} - 580x + 26080$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(228))$	8
Old forms	106
Cusp forms $S_{4}(\Gamma_0(228))$	114
Eisenstein series $E_{4}(\Gamma_0(228))$	12
Modular forms $M_{4}(\Gamma_0(228))$	126

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(228)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(76))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(114))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.176 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, 3, 4, 6, 12, 19, 38, 57, 76, 114, 228]: [1, 2, 2, 3, 5, 9, 6, 17, 22, 33, 64, 126] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 8 = 1 x 8 (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(228, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 228:4 1000

← 227.4.a · 229.4.a →