Atlas › Modular forms › Weight 12 › Level 5 › 5.12.a

Newspace 5.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(5))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
5.12.a.a	1	$\mathbb{Q}$	$q + 34q^{2} - 792q^{3} - 892q^{4} + 3125q^{5} - 26928q^{6} - 17556q^{7} + O(q^{8})$	−	$-1$
5.12.a.b	2	$\mathbb{Q}(\sqrt{151})$	$2q - 20q^{2} - 220q^{3} + 6976q^{4} - 6250q^{5} + 60184q^{6} + 57900q^{7} + O(q^{8})$	+	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
5.12.a.a	$x + 758$
5.12.a.b	$x^{2} + 240x - 203644$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(5))$	3
Old forms	2
Cusp forms $S_{12}(\Gamma_0(5))$	5
Eisenstein series $E_{12}(\Gamma_0(5))$	2
Modular forms $M_{12}(\Gamma_0(5))$	7

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(5)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 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, 5]: [2, 7] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (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(5, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 5:12 1000

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