Atlas › Modular forms › Weight 4 › Level 116 › 116.4.a

Newspace 116.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(116))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
116.4.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q - 10q^{5} - 20q^{7} + O(q^{8})$	+
116.4.a.b	2	$\mathbb{Q}(\sqrt{22})$	$2q + 10q^{3} + 30q^{5} + O(q^{8})$	−
116.4.a.c	3	degree 3	$3q - 10q^{3} - 20q^{5} + 8q^{7} + O(q^{8})$	−

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 $T$
116.4.a.a	$x^{2} + 10x + 12$
116.4.a.b	$x^{2} - 40x + 378$
116.4.a.c	$x^{3} + 30x^{2} + 66x - 36$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(116))$	7
Old forms	35
Cusp forms $S_{4}(\Gamma_0(116))$	42
Eisenstein series $E_{4}(\Gamma_0(116))$	6
Modular forms $M_{4}(\Gamma_0(116))$	48

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(116)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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, 29, 58, 116]: [1, 2, 3, 9, 25, 48] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(116, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 116:4 1000

← 115.4.a · 117.4.a →