Atlas › Modular forms › Weight 6 › Level 9 › 9.6.a

Newspace 9.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(9))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
9.6.a.a	1	$\mathbb{Q}$	$q + 6q^{2} + 4q^{4} - 6q^{5} - 40q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
9.6.a.a	$x + 12$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(9))$	1
Old forms	2
Cusp forms $S_{6}(\Gamma_0(9))$	3
Eisenstein series $E_{6}(\Gamma_0(9))$	4
Modular forms $M_{6}(\Gamma_0(9))$	7

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(9)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 3, 6] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; 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(9, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 9:6 1000

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