Atlas › Modular forms › Weight 2 › Level 216 › 216.2.a

Newspace 216.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(216))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
216.2.a.a	1	$\mathbb{Q}$	$q - 4q^{5} - 3q^{7} + O(q^{8})$	216a1 rank 1
216.2.a.b	1	$\mathbb{Q}$	$q - q^{5} + 3q^{7} + O(q^{8})$	216b1 rank 0
216.2.a.c	1	$\mathbb{Q}$	$q + q^{5} + 3q^{7} + O(q^{8})$	216c1 rank 0
216.2.a.d	1	$\mathbb{Q}$	$q + 4q^{5} - 3q^{7} + O(q^{8})$	216d1 rank 0

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 $T$
216.2.a.a	$x + 13$
216.2.a.b	$x - 8$
216.2.a.c	$x - 10$
216.2.a.d	$x + 5$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(216))$	4
Old forms	21
Cusp forms $S_{2}(\Gamma_0(216))$	25
Eisenstein series $E_{2}(\Gamma_0(216))$	23
Modular forms $M_{2}(\Gamma_0(216))$	48

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(216)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(72))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(108))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216]: [0, 1, 1, 2, 3, 3, 2, 5, 5, 8, 4, 9, 11, 16, 21, 40] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; 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(216, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 216:2 1000

← 215.2.a · 217.2.a →