Atlas › Modular forms › Weight 4 › Level 114 › 114.4.a

Newspace 114.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(114))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
114.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} - 19q^{5} + 6q^{6} + 9q^{7} + O(q^{8})$	+ + +	$+1$
114.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 7q^{5} - 6q^{6} - 15q^{7} + O(q^{8})$	+ − +	$-1$
114.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} + 12q^{5} - 6q^{6} + 4q^{7} + O(q^{8})$	+ − −	$+1$
114.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 11q^{5} - 6q^{6} - 15q^{7} + O(q^{8})$	− + +	$-1$
114.4.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q + 4q^{2} - 6q^{3} + 8q^{4} + 18q^{5} - 12q^{6} + 4q^{7} + O(q^{8})$	− + −	$+1$
114.4.a.f	2	$\mathbb{Q}(\sqrt{273})$	$2q + 4q^{2} + 6q^{3} + 8q^{4} + 11q^{5} + 12q^{6} + 9q^{7} + O(q^{8})$	− − +	$+1$

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 $T$
114.4.a.a	$x - 8$
114.4.a.b	$x + 52$
114.4.a.c	$x - 24$
114.4.a.d	$x + 56$
114.4.a.e	$x^{2} - 30x - 3600$
114.4.a.f	$x^{2} - 38x + 88$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(114))$	8
Old forms	48
Cusp forms $S_{4}(\Gamma_0(114))$	56
Eisenstein series $E_{4}(\Gamma_0(114))$	8
Modular forms $M_{4}(\Gamma_0(114))$	64

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(114)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.127 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, 6, 19, 38, 57, 114]: [1, 2, 2, 5, 6, 17, 22, 64] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 8 = 1 x 8 (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(114, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 114:4 1000

← 113.4.a · 115.4.a →