Atlas › Modular forms › Weight 4 › Level 27 › 27.4.a

Newspace 27.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(27))$

Newforms

Label	Dim	Coefficient field	Trace form
27.4.a.a	1	$\mathbb{Q}$	$q - 3q^{2} + q^{4} - 15q^{5} - 25q^{7} + O(q^{8})$
27.4.a.b	1	$\mathbb{Q}$	$q + 3q^{2} + q^{4} + 15q^{5} - 25q^{7} + O(q^{8})$
27.4.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 20q^{4} + 22q^{7} + O(q^{8})$

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 $T$
27.4.a.a	$x + 18$
27.4.a.b	$x - 18$
27.4.a.c	$x^{2} - 162$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(27))$	4
Old forms	2
Cusp forms $S_{4}(\Gamma_0(27))$	6
Eisenstein series $E_{4}(\Gamma_0(27))$	6
Modular forms $M_{4}(\Gamma_0(27))$	12

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.097 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]: [1, 2, 4, 10] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 27:4 1000

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