Atlas › Modular forms › Weight 8 › Level 54 › 54.8.a

Newspace 54.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(54))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
54.8.a.a	1	$\mathbb{Q}$	$q - 8q^{2} + 64q^{4} - 120q^{5} + 377q^{7} + O(q^{8})$	+
54.8.a.b	1	$\mathbb{Q}$	$q - 8q^{2} + 64q^{4} + 105q^{5} - 937q^{7} + O(q^{8})$	+
54.8.a.c	1	$\mathbb{Q}$	$q - 8q^{2} + 64q^{4} + 312q^{5} + 323q^{7} + O(q^{8})$	+
54.8.a.d	1	$\mathbb{Q}$	$q + 8q^{2} + 64q^{4} - 312q^{5} + 323q^{7} + O(q^{8})$	−
54.8.a.e	1	$\mathbb{Q}$	$q + 8q^{2} + 64q^{4} - 105q^{5} - 937q^{7} + O(q^{8})$	−
54.8.a.f	1	$\mathbb{Q}$	$q + 8q^{2} + 64q^{4} + 120q^{5} + 377q^{7} + O(q^{8})$	−
54.8.a.g	2	$\mathbb{Q}(\sqrt{329})$	$2q - 16q^{2} + 128q^{4} + 48q^{5} + 880q^{7} + O(q^{8})$	+
54.8.a.h	2	$\mathbb{Q}(\sqrt{329})$	$2q + 16q^{2} + 128q^{4} - 48q^{5} + 880q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
54.8.a.a	$x - 1011$
54.8.a.b	$x + 2706$
54.8.a.c	$x - 1281$
54.8.a.d	$x - 657$
54.8.a.e	$x + 2916$
54.8.a.f	$x - 1251$
54.8.a.g	$x^{2} - 2688x - 2031120$
54.8.a.h	$x^{2} - 2592x + 720252$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(54))$	10
Old forms	47
Cusp forms $S_{8}(\Gamma_0(54))$	57
Eisenstein series $E_{8}(\Gamma_0(54))$	12
Modular forms $M_{8}(\Gamma_0(54))$	69

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(54)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 6}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 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, 3, 6, 9, 18, 27, 54]: [1, 3, 3, 9, 8, 23, 22, 65] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 10 = 1 x 10 (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(54, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 54:8 1000

← 53.8.a · 55.8.a →