Atlas › Modular forms › Weight 10 › Level 9 › 9.10.a

Newspace 9.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(9))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
9.10.a.a	1	$\mathbb{Q}$	$q - 18q^{2} - 188q^{4} + 1530q^{5} + 9128q^{7} + O(q^{8})$
9.10.a.b	1	$\mathbb{Q}$	$q - 512q^{4} - 12580q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$
9.10.a.c	1	$\mathbb{Q}$	$q + 36q^{2} + 784q^{4} + 1314q^{5} - 4480q^{7} + O(q^{8})$

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 $T$
9.10.a.a	$x - 1512$
9.10.a.b	$x$
9.10.a.c	$x - 1350$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(9))$	3
Old forms	4
Cusp forms $S_{10}(\Gamma_0(9))$	7
Eisenstein series $E_{10}(\Gamma_0(9))$	4
Modular forms $M_{10}(\Gamma_0(9))$	11

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(9)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.29 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]: [1, 4, 10] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 3 = 1 x 3 (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, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 9:10 1000

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