Atlas › Modular forms › Weight 2 › Level 177 › 177.2.a

Newspace 177.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(177))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
177.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 2q^{3} + 3q^{4} - 6q^{5} - 3q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
177.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
177.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 2q^{3} - q^{4} + 2q^{5} + q^{6} + q^{7} + O(q^{8})$	− +	$+1$
177.2.a.d	3	degree 3	$3q - 3q^{3} + 2q^{4} - 2q^{5} + 9q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
177.2.a.a	$x^{2} + 9x + 19$
177.2.a.b	$x^{2} + x - 1$
177.2.a.c	$x^{2} - 3x + 1$
177.2.a.d	$x^{3} + 2x^{2} - 14x + 13$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(177))$	9
Old forms	10
Cusp forms $S_{2}(\Gamma_0(177))$	19
Eisenstein series $E_{2}(\Gamma_0(177))$	3
Modular forms $M_{2}(\Gamma_0(177))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(177)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.262 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, 3, 59, 177]: [0, 1, 6, 22] (certified at ell = 2147482481)
T = [(2, 1), (5, 1)]; h of degree 9 = 1 x 9 (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(177, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 177:2 1000

← 176.2.a · 178.2.a →