Atlas › Modular forms › Weight 8 › Level 12 › 12.8.a

Newspace 12.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(12))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
12.8.a.a	1	$\mathbb{Q}$	$q - 27q^{3} - 378q^{5} - 832q^{7} + O(q^{8})$	+
12.8.a.b	1	$\mathbb{Q}$	$q + 27q^{3} + 270q^{5} + 1112q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
12.8.a.a	$x + 1210$
12.8.a.b	$x - 1382$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(12))$	2
Old forms	9
Cusp forms $S_{8}(\Gamma_0(12))$	11
Eisenstein series $E_{8}(\Gamma_0(12))$	6
Modular forms $M_{8}(\Gamma_0(12))$	17

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 12]: [1, 3, 3, 5, 9, 17] (certified at ell = 2147483647)
T = [(5, 1), (7, 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(12, 8)

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

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

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