Atlas › Modular forms › Weight 4 › Level 22 › 22.4.a

Newspace 22.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(22))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
22.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 7q^{3} + 4q^{4} - 19q^{5} + 14q^{6} + 14q^{7} + O(q^{8})$	+ −	$-1$
22.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{3} + 4q^{4} + 14q^{5} - 8q^{6} - 8q^{7} + O(q^{8})$	+ +	$+1$
22.4.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 4q^{4} - 3q^{5} + 2q^{6} - 10q^{7} + O(q^{8})$	− −	$+1$

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 $T$
22.4.a.a	$x + 26$
22.4.a.b	$x - 18$
22.4.a.c	$x + 2$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(22))$	3
Old forms	4
Cusp forms $S_{4}(\Gamma_0(22))$	7
Eisenstein series $E_{4}(\Gamma_0(22))$	4
Modular forms $M_{4}(\Gamma_0(22))$	11

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(22)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.099 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, 11, 22]: [1, 2, 4, 11] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 3 = 1 x 3 (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(22, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 22:4 1000

← 21.4.a · 23.4.a →