Atlas › Modular forms › Weight 2 › Level 188 › 188.2.a

Newspace 188.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(188))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
188.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{3} - 2q^{5} - 7q^{7} + O(q^{8})$	−
188.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{3} + 5q^{7} + O(q^{8})$	+

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 $T$
188.2.a.a	$x^{2} + 5x + 5$
188.2.a.b	$x^{2} - x - 3$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(188))$	4
Old forms	18
Cusp forms $S_{2}(\Gamma_0(188))$	22
Eisenstein series $E_{2}(\Gamma_0(188))$	5
Modular forms $M_{2}(\Gamma_0(188))$	27

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(188)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.202 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, 47, 94, 188]: [0, 1, 2, 5, 14, 27] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; 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(188, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 188:2 1000

← 187.2.a · 189.2.a →