Atlas › Modular forms › Weight 4 › Level 188 › 188.4.a

Newspace 188.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(188))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
188.4.a.a	6	degree 6	$6q - 4q^{3} - 10q^{5} - 44q^{7} + O(q^{8})$	+
188.4.a.b	6	degree 6	$6q + 8q^{3} + 40q^{7} + O(q^{8})$	−

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 $T$
188.4.a.a	$x^{6} + 14x^{5} - 123x^{4} - 2126x^{3} + 1727x^{2} + 80168x + 132970$
188.4.a.b	$x^{6} - 8x^{5} - 821x^{4} + 6740x^{3} + 161069x^{2} - 816058x - 10480872$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(188))$	12
Old forms	57
Cusp forms $S_{4}(\Gamma_0(188))$	69
Eisenstein series $E_{4}(\Gamma_0(188))$	6
Modular forms $M_{4}(\Gamma_0(188))$	75

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(188)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.125 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]: [1, 2, 3, 13, 38, 75] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 188:4 1000

← 187.4.a · 189.4.a →