Atlas › Modular forms › Weight 10 › Level 16 › 16.10.a

Newspace 16.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(16))$

Newforms

Label	Dim	Coefficient field	Trace form
16.10.a.a	1	$\mathbb{Q}$	$q - 228q^{3} - 666q^{5} + 6328q^{7} + O(q^{8})$
16.10.a.b	1	$\mathbb{Q}$	$q - 68q^{3} + 1510q^{5} - 10248q^{7} + O(q^{8})$
16.10.a.c	1	$\mathbb{Q}$	$q + 60q^{3} - 2074q^{5} + 4344q^{7} + O(q^{8})$
16.10.a.d	1	$\mathbb{Q}$	$q + 156q^{3} + 870q^{5} + 952q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
16.10.a.a	$x + 894$
16.10.a.b	$x - 1442$
16.10.a.c	$x + 2014$
16.10.a.d	$x - 1026$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(16))$	4
Old forms	11
Cusp forms $S_{10}(\Gamma_0(16))$	15
Eisenstein series $E_{10}(\Gamma_0(16))$	6
Modular forms $M_{10}(\Gamma_0(16))$	21

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(16)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 4}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 3}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.296 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, 8, 16]: [1, 3, 6, 11, 20] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (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(16, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 16:10 1000

← 15.10.a · 17.10.a →