Atlas › Modular forms › Weight 6 › Level 36 › 36.6.a

Newspace 36.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(36))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
36.6.a.a	1	$\mathbb{Q}$	$q - 54q^{5} - 88q^{7} + O(q^{8})$
36.6.a.b	1	$\mathbb{Q}$	$q + 236q^{7} + O(q^{8})$			$\mathbb{Q}(\sqrt{-3})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
36.6.a.a	$x + 142$
36.6.a.b	$x - 236$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(36))$	2
Old forms	22
Cusp forms $S_{6}(\Gamma_0(36))$	24
Eisenstein series $E_{6}(\Gamma_0(36))$	12
Modular forms $M_{6}(\Gamma_0(36))$	36

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(36)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.12 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, 9, 12, 18, 36]: [1, 2, 3, 4, 7, 6, 13, 17, 33] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 2 = 1 x 2 (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(36, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 36:6 1000

← 35.6.a · 37.6.a →