Atlas › Modular forms › Weight 12 › Level 8 › 8.12.a

Newspace 8.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(8))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
8.12.a.a	1	$\mathbb{Q}$	$q - 36q^{3} - 3490q^{5} - 55464q^{7} + O(q^{8})$
8.12.a.b	2	$\mathbb{Q}(\sqrt{109})$	$2q + 56q^{3} + 7868q^{5} + 91056q^{7} + O(q^{8})$

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 $T$
8.12.a.a	$x + 3526$
8.12.a.b	$x^{2} - 7924x - 38324700$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(8))$	3
Old forms	6
Cusp forms $S_{12}(\Gamma_0(8))$	9
Eisenstein series $E_{12}(\Gamma_0(8))$	4
Modular forms $M_{12}(\Gamma_0(8))$	13

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(8)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.311 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]: [2, 4, 7, 13] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 2 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, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 8:12 1000

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