Atlas › Modular forms › Weight 12 › Level 22 › 22.12.a

Newspace 22.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(22))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
22.12.a.a	2	$\mathbb{Q}(\sqrt{331})$	$2q + 64q^{2} - 426q^{3} + 2048q^{4} + 2290q^{5} - 13632q^{6} - 86324q^{7} + O(q^{8})$	− +	$-1$
22.12.a.b	3	degree 3	$3q - 96q^{2} - 70q^{3} + 3072q^{4} - 5624q^{5} + 2240q^{6} - 30576q^{7} + O(q^{8})$	+ −	$-1$
22.12.a.c	3	degree 3	$3q - 96q^{2} + 106q^{3} + 3072q^{4} + 17344q^{5} - 3392q^{6} + 13204q^{7} + O(q^{8})$	+ +	$+1$
22.12.a.d	3	degree 3	$3q + 96q^{2} + 370q^{3} + 3072q^{4} - 1928q^{5} + 11840q^{6} + 58472q^{7} + O(q^{8})$	− −	$+1$

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 $T$
22.12.a.a	$x^{2} - 1864x - 45926832$
22.12.a.b	$x^{3} + 5694x^{2} + 5500220x - 633213048$
22.12.a.c	$x^{3} - 17450x^{2} - 14484096x + 957692196576$
22.12.a.d	$x^{3} + 1558x^{2} - 95149604x - 215164247832$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(22))$	11
Old forms	20
Cusp forms $S_{12}(\Gamma_0(22))$	31
Eisenstein series $E_{12}(\Gamma_0(22))$	4
Modular forms $M_{12}(\Gamma_0(22))$	35

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(22)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.359 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]: [2, 4, 12, 35] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 5 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, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 22:12 1000

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