Atlas › Modular forms › Weight 4 › Level 12 › 12.4.a

Newspace 12.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(12))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
12.4.a.a	1	$\mathbb{Q}$	$q + 3q^{3} - 18q^{5} + 8q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
12.4.a.a	$x - 6$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(12))$	1
Old forms	2
Cusp forms $S_{4}(\Gamma_0(12))$	3
Eisenstein series $E_{4}(\Gamma_0(12))$	6
Modular forms $M_{4}(\Gamma_0(12))$	9

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 2, 2, 3, 5, 9] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; 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(12, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 12:4 1000

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