Atlas › Modular forms › Weight 8 › Level 27 › 27.8.a

Newspace 27.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(27))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
27.8.a.a	1	$\mathbb{Q}$	$q - 128q^{4} + 1763q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$
27.8.a.b	2	$\mathbb{Q}(\sqrt{65})$	$2q - 9q^{2} + 77q^{4} - 180q^{5} + 700q^{7} + O(q^{8})$
27.8.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 40q^{4} - 1118q^{7} + O(q^{8})$
27.8.a.d	2	$\mathbb{Q}(\sqrt{42})$	$2q + 500q^{4} - 2522q^{7} + O(q^{8})$
27.8.a.e	2	$\mathbb{Q}(\sqrt{65})$	$2q + 9q^{2} + 77q^{4} + 180q^{5} + 700q^{7} + O(q^{8})$

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 $T$
27.8.a.a	$x$
27.8.a.b	$x^{2} + 189x + 7614$
27.8.a.c	$x^{2} - 117612$
27.8.a.d	$x^{2} - 166698$
27.8.a.e	$x^{2} - 189x + 7614$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(27))$	9
Old forms	9
Cusp forms $S_{8}(\Gamma_0(27))$	18
Eisenstein series $E_{8}(\Gamma_0(27))$	6
Modular forms $M_{8}(\Gamma_0(27))$	24

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(27)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 3, 8, 22] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 9 = 1 x 9 (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(27, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 27:8 1000

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