Atlas › Modular forms › Weight 4 › Level 25 › 25.4.a

Newspace 25.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(25))$

Newforms

Label	Dim	Coefficient field	Trace form
25.4.a.a	1	$\mathbb{Q}$	$q - q^{2} - 7q^{3} - 7q^{4} + 7q^{6} - 6q^{7} + O(q^{8})$
25.4.a.b	1	$\mathbb{Q}$	$q + q^{2} + 7q^{3} - 7q^{4} + 7q^{6} + 6q^{7} + O(q^{8})$
25.4.a.c	1	$\mathbb{Q}$	$q + 4q^{2} - 2q^{3} + 8q^{4} - 8q^{6} - 6q^{7} + O(q^{8})$

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 $T$
25.4.a.a	$x + 8$
25.4.a.b	$x - 8$
25.4.a.c	$x - 2$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(25))$	3
Old forms	2
Cusp forms $S_{4}(\Gamma_0(25))$	5
Eisenstein series $E_{4}(\Gamma_0(25))$	6
Modular forms $M_{4}(\Gamma_0(25))$	11

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.097 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]: [1, 3, 9] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (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(25, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 25:4 1000

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