Atlas › Modular forms › Weight 2 › Level 222 › 222.2.a

Newspace 222.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(222))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
222.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 4q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$	222e1 rank 0
222.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} + O(q^{8})$	+ + −	$+1$	222c1 rank 0
222.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 4q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − +	$+1$	222d1 rank 0
222.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} + 3q^{7} + O(q^{8})$	− + +	$+1$	222b1 rank 0
222.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + O(q^{8})$	− − −	$+1$	222a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
222.2.a.a	$x - 65$
222.2.a.b	$x - 44$
222.2.a.c	$x - 39$
222.2.a.d	$x + 23$
222.2.a.e	$x + 33$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(222))$	5
Old forms	30
Cusp forms $S_{2}(\Gamma_0(222))$	35
Eisenstein series $E_{2}(\Gamma_0(222))$	7
Modular forms $M_{2}(\Gamma_0(222))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(222)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(74))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(111))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 6, 37, 74, 111, 222]: [0, 1, 1, 3, 3, 11, 14, 42] (certified at ell = 2147483629)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 5 = 1 x 5 (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(222, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 222:2 1000

← 221.2.a · 223.2.a →