Atlas › Modular forms › Weight 6 › Level 8 › 8.6.a

Newspace 8.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(8))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
8.6.a.a	1	$\mathbb{Q}$	$q + 20q^{3} - 74q^{5} - 24q^{7} + O(q^{8})$

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 $T$
8.6.a.a	$x + 54$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(8))$	1
Old forms	2
Cusp forms $S_{6}(\Gamma_0(8))$	3
Eisenstein series $E_{6}(\Gamma_0(8))$	4
Modular forms $M_{6}(\Gamma_0(8))$	7

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(8)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 2, 4, 7] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 1 = 1 x 1 (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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 8:6 1000

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