Atlas › Modular forms › Weight 8 › Level 22 › 22.8.a

Newspace 22.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(22))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
22.8.a.a	1	$\mathbb{Q}$	$q - 8q^{2} - 19q^{3} + 64q^{4} + 317q^{5} + 152q^{6} - 1030q^{7} + O(q^{8})$	+ −	$-1$
22.8.a.b	1	$\mathbb{Q}$	$q - 8q^{2} + 91q^{3} + 64q^{4} + 185q^{5} - 728q^{6} - 722q^{7} + O(q^{8})$	+ +	$+1$
22.8.a.c	1	$\mathbb{Q}$	$q + 8q^{2} - 21q^{3} + 64q^{4} - 551q^{5} - 168q^{6} + 62q^{7} + O(q^{8})$	− +	$-1$
22.8.a.d	2	$\mathbb{Q}(\sqrt{14881})$	$2q + 16q^{2} - 23q^{3} + 128q^{4} + 331q^{5} - 184q^{6} + 1794q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
22.8.a.a	$x - 932$
22.8.a.b	$x - 646$
22.8.a.c	$x + 1674$
22.8.a.d	$x^{2} - 970x + 220344$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(22))$	5
Old forms	14
Cusp forms $S_{8}(\Gamma_0(22))$	19
Eisenstein series $E_{8}(\Gamma_0(22))$	4
Modular forms $M_{8}(\Gamma_0(22))$	23

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(22)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.265 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, 3, 8, 23] (certified at ell = 2147483171)
T = [(3, 1), (5, 3)]; 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, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 22:8 1000

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