Atlas › Modular forms › Weight 12 › Level 3 › 3.12.a

Newspace 3.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(3))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
3.12.a.a	1	$\mathbb{Q}$	$q + 78q^{2} - 243q^{3} + 4036q^{4} - 5370q^{5} - 18954q^{6} - 27760q^{7} + O(q^{8})$	+	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
3.12.a.a	$x + 5292$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(3))$	1
Old forms	2
Cusp forms $S_{12}(\Gamma_0(3))$	3
Eisenstein series $E_{12}(\Gamma_0(3))$	2
Modular forms $M_{12}(\Gamma_0(3))$	5

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 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, 3]: [2, 5] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; 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(3, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 3:12 1000

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