Atlas › Modular forms › Weight 8 › Level 48 › 48.8.a

Newspace 48.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(48))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
48.8.a.a	1	$\mathbb{Q}$	$q - 27q^{3} - 530q^{5} - 120q^{7} + O(q^{8})$	+
48.8.a.b	1	$\mathbb{Q}$	$q - 27q^{3} - 114q^{5} + 1576q^{7} + O(q^{8})$	+
48.8.a.c	1	$\mathbb{Q}$	$q - 27q^{3} + 110q^{5} - 504q^{7} + O(q^{8})$	+
48.8.a.d	1	$\mathbb{Q}$	$q - 27q^{3} + 270q^{5} - 1112q^{7} + O(q^{8})$	+
48.8.a.e	1	$\mathbb{Q}$	$q + 27q^{3} - 378q^{5} + 832q^{7} + O(q^{8})$	−
48.8.a.f	1	$\mathbb{Q}$	$q + 27q^{3} - 26q^{5} - 1056q^{7} + O(q^{8})$	−
48.8.a.g	1	$\mathbb{Q}$	$q + 27q^{3} + 390q^{5} + 64q^{7} + O(q^{8})$	−

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 $T$
48.8.a.a	$x + 890$
48.8.a.b	$x - 4614$
48.8.a.c	$x + 1402$
48.8.a.d	$x + 3066$
48.8.a.e	$x - 2118$
48.8.a.f	$x + 3194$
48.8.a.g	$x - 582$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(48))$	7
Old forms	43
Cusp forms $S_{8}(\Gamma_0(48))$	50
Eisenstein series $E_{8}(\Gamma_0(48))$	12
Modular forms $M_{8}(\Gamma_0(48))$	62

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(48)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 8}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 5}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.328 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, 4, 6, 8, 12, 16, 24, 48]: [1, 3, 3, 5, 9, 9, 17, 16, 32, 60] (certified at ell = 2147483629)
T = [(5, 1), (7, 3)]; h of degree 7 = 1 x 7 (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(48, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 48:8 1000

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