Atlas › Modular forms › Weight 2 › Level 117 › 117.2.a

Newspace 117.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(117))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
117.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	−	117a1 rank 1
117.2.a.b	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} + 4q^{7} + O(q^{8})$	−
117.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
117.2.a.a	$x + 3$
117.2.a.b	$x^{2} - 3$
117.2.a.c	$x^{2} - 2x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(117))$	5
Old forms	6
Cusp forms $S_{2}(\Gamma_0(117))$	11
Eisenstein series $E_{2}(\Gamma_0(117))$	7
Modular forms $M_{2}(\Gamma_0(117))$	18

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(117)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.196 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, 3, 9, 13, 39, 117]: [0, 1, 2, 1, 6, 16] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 5 = 1 x 5 (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(117, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 117:2 1000

← 116.2.a · 118.2.a →