Atlas › Modular forms › Weight 8 › Level 20 › 20.8.a

Newspace 20.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(20))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
20.8.a.a	1	$\mathbb{Q}$	$q - 6q^{3} - 125q^{5} - 706q^{7} + O(q^{8})$	+
20.8.a.b	2	$\mathbb{Q}(\sqrt{1129})$	$2q - 20q^{3} + 250q^{5} + 1660q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
20.8.a.a	$x + 712$
20.8.a.b	$x^{2} - 1640x + 383376$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(20))$	3
Old forms	15
Cusp forms $S_{8}(\Gamma_0(20))$	18
Eisenstein series $E_{8}(\Gamma_0(20))$	6
Modular forms $M_{8}(\Gamma_0(20))$	24

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(20)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.261 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, 4, 5, 10, 20]: [1, 3, 5, 5, 13, 24] (certified at ell = 2147483629)
T = [(3, 1), (7, 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(20, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 20:8 1000

← 19.8.a · 21.8.a →