Atlas › Modular forms › Weight 6 › Level 66 › 66.6.a

Newspace 66.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(66))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
66.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 9q^{3} + 16q^{4} - 14q^{5} + 36q^{6} - 112q^{7} + O(q^{8})$	+ + −	$+1$
66.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} - 9q^{3} + 16q^{4} - 14q^{5} + 36q^{6} + 130q^{7} + O(q^{8})$	+ + +	$-1$
66.6.a.c	1	$\mathbb{Q}$	$q - 4q^{2} + 9q^{3} + 16q^{4} - 14q^{5} - 36q^{6} - 130q^{7} + O(q^{8})$	+ − −	$-1$
66.6.a.d	1	$\mathbb{Q}$	$q + 4q^{2} - 9q^{3} + 16q^{4} + 50q^{5} - 36q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$
66.6.a.e	2	$\mathbb{Q}(\sqrt{8761})$	$2q - 8q^{2} + 18q^{3} + 32q^{4} - 14q^{5} - 72q^{6} + 210q^{7} + O(q^{8})$	+ − +	$+1$
66.6.a.f	2	$\mathbb{Q}(\sqrt{2161})$	$2q + 8q^{2} + 18q^{3} + 32q^{4} + 50q^{5} + 72q^{6} + 96q^{7} + O(q^{8})$	− − −	$+1$

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 $T$
66.6.a.a	$x + 350$
66.6.a.b	$x - 376$
66.6.a.c	$x + 404$
66.6.a.d	$x - 56$
66.6.a.e	$x^{2} - 616x + 59820$
66.6.a.f	$x^{2} - 338x - 25464$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(66))$	8
Old forms	48
Cusp forms $S_{6}(\Gamma_0(66))$	56
Eisenstein series $E_{6}(\Gamma_0(66))$	8
Modular forms $M_{6}(\Gamma_0(66))$	64

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(66)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.279 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, 6, 11, 22, 33, 66]: [1, 2, 3, 7, 6, 17, 22, 64] (certified at ell = 2147483171)
T = [(5, 1), (7, 3)]; h of degree 8 = 1 x 8 (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(66, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 66:6 1000

← 65.6.a · 67.6.a →