Atlas › Modular forms › Weight 10 › Level 22 › 22.10.a

Newspace 22.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(22))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
22.10.a.a	1	$\mathbb{Q}$	$q + 16q^{2} - 41q^{3} + 256q^{4} - 1039q^{5} - 656q^{6} - 3482q^{7} + O(q^{8})$	− −	$-1$
22.10.a.b	1	$\mathbb{Q}$	$q + 16q^{2} + 137q^{3} + 256q^{4} - 595q^{5} + 2192q^{6} + 11354q^{7} + O(q^{8})$	− +	$+1$
22.10.a.c	1	$\mathbb{Q}$	$q + 16q^{2} + 201q^{3} + 256q^{4} + 2349q^{5} + 3216q^{6} - 8806q^{7} + O(q^{8})$	− +	$+1$
22.10.a.d	2	$\mathbb{Q}(\sqrt{889})$	$2q - 32q^{2} - 21q^{3} + 512q^{4} - 521q^{5} + 336q^{6} - 7490q^{7} + O(q^{8})$	+ −	$+1$
22.10.a.e	2	$\mathbb{Q}(\sqrt{463})$	$2q - 32q^{2} + 34q^{3} + 512q^{4} - 1478q^{5} - 544q^{6} + 8196q^{7} + O(q^{8})$	+ +	$-1$

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 $T$
22.10.a.a	$x + 1080$
22.10.a.b	$x + 458$
22.10.a.c	$x - 2550$
22.10.a.d	$x^{2} + 542x - 2423760$
22.10.a.e	$x^{2} + 1444x - 1878908$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(22))$	7
Old forms	18
Cusp forms $S_{10}(\Gamma_0(22))$	25
Eisenstein series $E_{10}(\Gamma_0(22))$	4
Modular forms $M_{10}(\Gamma_0(22))$	29

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(22)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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, 10, 29] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 3 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, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 22:10 1000

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