Atlas › Modular forms › Weight 2 › Level 228 › 228.2.a

Newspace 228.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(228))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
228.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + q^{7} + O(q^{8})$	+ −	228b1 rank 1
228.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + O(q^{8})$	+ +	228a1 rank 0
228.2.a.c	2	$\mathbb{Q}(\sqrt{33})$	$2q + 2q^{3} + 3q^{5} + q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{23}$ 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 $T$
228.2.a.a	$x + 29$
228.2.a.b	$x - 72$
228.2.a.c	$x^{2} + 49x - 68$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(228))$	4
Old forms	31
Cusp forms $S_{2}(\Gamma_0(228))$	35
Eisenstein series $E_{2}(\Gamma_0(228))$	11
Modular forms $M_{2}(\Gamma_0(228))$	46

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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]: [0, 1, 1, 2, 3, 5, 2, 7, 8, 13, 24, 46] (certified at ell = 2147483647)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (23, 9)]; h of degree 4 = 1 x 4 (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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 228:2 1000

← 227.2.a · 229.2.a →