Atlas › Modular forms › Weight 8 › Level 8 › 8.8.a

Newspace 8.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(8))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
8.8.a.a	1	$\mathbb{Q}$	$q - 84q^{3} - 82q^{5} - 456q^{7} + O(q^{8})$
8.8.a.b	1	$\mathbb{Q}$	$q + 44q^{3} + 430q^{5} - 1224q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
8.8.a.a	$x + 166$
8.8.a.b	$x - 474$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(8))$	2
Old forms	3
Cusp forms $S_{8}(\Gamma_0(8))$	5
Eisenstein series $E_{8}(\Gamma_0(8))$	4
Modular forms $M_{8}(\Gamma_0(8))$	9

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 4, 8]: [1, 3, 5, 9] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 2 = 1 x 2 (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(8, 8)

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

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

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