Atlas › Modular forms › Weight 12 › Level 24 › 24.12.a

Newspace 24.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(24))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
24.12.a.a	1	$\mathbb{Q}$	$q - 243q^{3} - 7130q^{5} - 19536q^{7} + O(q^{8})$	+
24.12.a.b	1	$\mathbb{Q}$	$q - 243q^{3} + 1190q^{5} + 18480q^{7} + O(q^{8})$	+
24.12.a.c	1	$\mathbb{Q}$	$q + 243q^{3} + 1870q^{5} - 72312q^{7} + O(q^{8})$	−
24.12.a.d	2	$\mathbb{Q}(\sqrt{3061})$	$2q + 486q^{3} + 1564q^{5} + 37776q^{7} + O(q^{8})$	−

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 $T$
24.12.a.a	$x + 65738$
24.12.a.b	$x - 56630$
24.12.a.c	$x + 215066$
24.12.a.d	$x^{2} - 114892x + 2848680100$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(24))$	5
Old forms	35
Cusp forms $S_{12}(\Gamma_0(24))$	40
Eisenstein series $E_{12}(\Gamma_0(24))$	8
Modular forms $M_{12}(\Gamma_0(24))$	48

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(24)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 8}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 3}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.383 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, 8, 12, 24]: [2, 4, 5, 7, 13, 13, 25, 48] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 3 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(24, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 24:12 1000

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