Atlas › Modular forms › Weight 12 › Level 6 › 6.12.a

Newspace 6.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(6))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
6.12.a.a	1	$\mathbb{Q}$	$q - 32q^{2} - 243q^{3} + 1024q^{4} + 5766q^{5} + 7776q^{6} + 72464q^{7} + O(q^{8})$	+ +	$+1$
6.12.a.b	1	$\mathbb{Q}$	$q - 32q^{2} + 243q^{3} + 1024q^{4} - 11730q^{5} - 7776q^{6} - 50008q^{7} + O(q^{8})$	+ −	$-1$
6.12.a.c	1	$\mathbb{Q}$	$q + 32q^{2} + 243q^{3} + 1024q^{4} + 3630q^{5} + 7776q^{6} + 32936q^{7} + O(q^{8})$	− −	$+1$

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 $T$
6.12.a.a	$x - 78230$
6.12.a.b	$x + 61738$
6.12.a.c	$x - 36566$

Dimensions

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

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(6)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 6]: [2, 4, 5, 13] (certified at ell = 2147483647)
T = [(5, 1), (7, 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(6, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 6:12 1000

← 5.12.a · 7.12.a →