Atlas › Modular forms › Weight 2 › Level 125 › 125.2.a

Newspace 125.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(125))$

Newforms

Label	Dim	Coefficient field	Trace form
125.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 3q^{3} - q^{4} - q^{6} - 6q^{7} + O(q^{8})$
125.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 3q^{3} - q^{4} - q^{6} + 6q^{7} + O(q^{8})$
125.2.a.c	4	degree 4	$4q + 8q^{4} - 2q^{6} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
125.2.a.a	$x^{2} + 10x + 20$
125.2.a.b	$x^{2} - 10x + 20$
125.2.a.c	$x^{4} - 65x^{2} + 275$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(125))$	8
Old forms	0
Cusp forms $S_{2}(\Gamma_0(125))$	8
Eisenstein series $E_{2}(\Gamma_0(125))$	9
Modular forms $M_{2}(\Gamma_0(125))$	17

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.175 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, 5, 25, 125]: [0, 1, 3, 13] (certified at ell = 2147482801)
T = [(2, 1), (3, 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(125, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 125:2 1000

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