Atlas › Modular forms › Weight 4 › Level 66 › 66.4.a

Newspace 66.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(66))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
66.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{6} + 14q^{7} + O(q^{8})$	+ − −	$+1$
66.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} + 10q^{5} - 6q^{6} + 16q^{7} + O(q^{8})$	− + −	$+1$
66.4.a.c	2	$\mathbb{Q}(\sqrt{97})$	$2q + 4q^{2} + 6q^{3} + 8q^{4} + 10q^{5} + 12q^{6} - 2q^{7} + O(q^{8})$	− − +	$+1$

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 $T$
66.4.a.a	$x - 14$
66.4.a.b	$x - 26$
66.4.a.c	$x^{2} - 8x - 372$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(66))$	4
Old forms	28
Cusp forms $S_{4}(\Gamma_0(66))$	32
Eisenstein series $E_{4}(\Gamma_0(66))$	8
Modular forms $M_{4}(\Gamma_0(66))$	40

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 2, 5, 4, 11, 14, 40] (certified at ell = 2147483171)
T = [(5, 1), (7, 1)]; h of degree 4 = 1 x 4 (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(66, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 66:4 1000

← 65.4.a · 67.4.a →