Atlas › Modular forms › Weight 4 › Level 125 › 125.4.a

Newspace 125.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(125))$

Newforms

Label	Dim	Coefficient field	Trace form
125.4.a.a	4	degree 4	$4q - 8q^{4} - 22q^{6} + O(q^{8})$
125.4.a.b	6	degree 6	$6q - 7q^{2} - 14q^{3} + 33q^{4} - 13q^{6} - 67q^{7} + O(q^{8})$
125.4.a.c	6	degree 6	$6q + 7q^{2} + 14q^{3} + 33q^{4} - 13q^{6} + 67q^{7} + O(q^{8})$
125.4.a.d	8	degree 8	$8q + 44q^{4} + 46q^{6} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
125.4.a.a	$x^{4} - 13x^{2} + 31$
125.4.a.b	$x^{6} + 21x^{5} + 66x^{4} - 603x^{3} - 1351x^{2} + 3506x - 1676$
125.4.a.c	$x^{6} - 21x^{5} + 66x^{4} + 603x^{3} - 1351x^{2} - 3506x - 1676$
125.4.a.d	$x^{8} - 256x^{6} + 24291x^{4} - 1011046x^{2} + 15550931$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(125))$	24
Old forms	9
Cusp forms $S_{4}(\Gamma_0(125))$	33
Eisenstein series $E_{4}(\Gamma_0(125))$	10
Modular forms $M_{4}(\Gamma_0(125))$	43

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(125)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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]: [1, 3, 9, 39] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 4 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 125:4 1000

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