Atlas › Modular forms › Weight 12 › Level 9 › 9.12.a

Newspace 9.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(9))$

Newforms

Label	Dim	Coefficient field	Trace form
9.12.a.a	1	$\mathbb{Q}$	$q - 78q^{2} + 4036q^{4} + 5370q^{5} - 27760q^{7} + O(q^{8})$
9.12.a.b	1	$\mathbb{Q}$	$q + 24q^{2} - 1472q^{4} - 4830q^{5} - 16744q^{7} + O(q^{8})$
9.12.a.c	2	$\mathbb{Q}(\sqrt{70})$	$2q + 944q^{4} + 116200q^{7} + O(q^{8})$

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 $T$
9.12.a.a	$x - 5292$
9.12.a.b	$x + 4806$
9.12.a.c	$x^{2} - 127575000$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(9))$	4
Old forms	5
Cusp forms $S_{12}(\Gamma_0(9))$	9
Eisenstein series $E_{12}(\Gamma_0(9))$	4
Modular forms $M_{12}(\Gamma_0(9))$	13

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(9)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 3}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 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, 9]: [2, 5, 12] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 4 = 1 x 4 (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(9, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 9:12 1000

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