Atlas › Modular forms › Weight 2 › Level 112 › 112.2.a

Newspace 112.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(112))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
112.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 4q^{5} - q^{7} + O(q^{8})$	+	112a1 rank 1
112.2.a.b	1	$\mathbb{Q}$	$q + 2q^{5} + q^{7} + O(q^{8})$	−	112b1 rank 0
112.2.a.c	1	$\mathbb{Q}$	$q + 2q^{3} - q^{7} + O(q^{8})$	+	112c1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
112.2.a.a	$x + 32$
112.2.a.b	$x + 14$
112.2.a.c	$x - 28$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(112))$	3
Old forms	8
Cusp forms $S_{2}(\Gamma_0(112))$	11
Eisenstein series $E_{2}(\Gamma_0(112))$	11
Modular forms $M_{2}(\Gamma_0(112))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(112)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 7, 8, 14, 16, 28, 56, 112]: [0, 1, 2, 1, 3, 4, 4, 7, 12, 20] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 3 = 1 x 3 (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(112, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 112:2 1000

← 111.2.a · 113.2.a →