Atlas › Modular forms › Weight 6 › Level 22 › 22.6.a

Newspace 22.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(22))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
22.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 21q^{3} + 16q^{4} + 81q^{5} + 84q^{6} + 98q^{7} + O(q^{8})$	+ −	$+1$
22.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} + q^{3} + 16q^{4} - 51q^{5} - 4q^{6} - 166q^{7} + O(q^{8})$	+ +	$-1$
22.6.a.c	1	$\mathbb{Q}$	$q + 4q^{2} - 29q^{3} + 16q^{4} - 31q^{5} - 116q^{6} - 230q^{7} + O(q^{8})$	− −	$-1$
22.6.a.d	2	$\mathbb{Q}(\sqrt{793})$	$2q + 8q^{2} + 29q^{3} + 32q^{4} - 13q^{5} + 116q^{6} - 14q^{7} + O(q^{8})$	− +	$+1$

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 $T$
22.6.a.a	$x - 60$
22.6.a.b	$x + 50$
22.6.a.c	$x + 60$
22.6.a.d	$x^{2} - 16x - 3108$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(22))$	5
Old forms	8
Cusp forms $S_{6}(\Gamma_0(22))$	13
Eisenstein series $E_{6}(\Gamma_0(22))$	4
Modular forms $M_{6}(\Gamma_0(22))$	17

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(22)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 6, 17] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; 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(22, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 22:6 1000

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