Atlas › Modular forms › Weight 12 › Level 27 › 27.12.a

Newspace 27.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(27))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
27.12.a.a	1	$\mathbb{Q}$	$q - 2048q^{4} + 77153q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$
27.12.a.b	2	$\mathbb{Q}(\sqrt{93})$	$2q + 2600q^{4} - 100142q^{7} + O(q^{8})$
27.12.a.c	4	degree 4	$4q - 9q^{2} + 6109q^{4} - 12060q^{5} - 12076q^{7} + O(q^{8})$
27.12.a.d	4	degree 4	$4q + 1636q^{4} + 28460q^{7} + O(q^{8})$
27.12.a.e	4	degree 4	$4q + 9q^{2} + 6109q^{4} + 12060q^{5} - 12076q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
27.12.a.a	$x$
27.12.a.b	$x^{2} - 271188$
27.12.a.c	$x^{4} + 12069x^{3} - 75287475x^{2} - 803711510325x + 857408118709122$
27.12.a.d	$x^{4} - 169304418x^{2} + 757130064694272$
27.12.a.e	$x^{4} - 12069x^{3} - 75287475x^{2} + 803711510325x + 857408118709122$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(27))$	15
Old forms	15
Cusp forms $S_{12}(\Gamma_0(27))$	30
Eisenstein series $E_{12}(\Gamma_0(27))$	6
Modular forms $M_{12}(\Gamma_0(27))$	36

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(27)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 3}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.546 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, 3, 9, 27]: [2, 5, 12, 34] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 7 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(27, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 27:12 1000

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