Atlas › Modular forms › Weight 12 › Level 12 › 12.12.a

Newspace 12.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(12))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
12.12.a.a	1	$\mathbb{Q}$	$q - 243q^{3} + 9990q^{5} - 86128q^{7} + O(q^{8})$	+
12.12.a.b	1	$\mathbb{Q}$	$q + 243q^{3} + 2862q^{5} + 9128q^{7} + O(q^{8})$	−

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 $T$
12.12.a.a	$x + 76138$
12.12.a.b	$x - 11990$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(12))$	2
Old forms	17
Cusp forms $S_{12}(\Gamma_0(12))$	19
Eisenstein series $E_{12}(\Gamma_0(12))$	6
Modular forms $M_{12}(\Gamma_0(12))$	25

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.338 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]: [2, 4, 5, 7, 13, 25] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 2 = 1 x 2 (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(12, 12)

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

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

← 11.12.a · 13.12.a →