Atlas › Modular forms › Weight 8 › Level 25 › 25.8.a

Newspace 25.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(25))$

Newforms

Label	Dim	Coefficient field	Trace form
25.8.a.a	1	$\mathbb{Q}$	$q + 14q^{2} + 48q^{3} + 68q^{4} + 672q^{6} + 1644q^{7} + O(q^{8})$
25.8.a.b	2	$\mathbb{Q}(\sqrt{19})$	$2q - 20q^{2} - 20q^{3} + 96q^{4} - 1016q^{6} + 100q^{7} + O(q^{8})$
25.8.a.c	2	$\mathbb{Q}(\sqrt{649})$	$2q - 15q^{2} - 40q^{3} + 181q^{4} + 949q^{6} + 600q^{7} + O(q^{8})$
25.8.a.d	2	$\mathbb{Q}(\sqrt{29})$	$2q - 24q^{4} - 696q^{6} + O(q^{8})$
25.8.a.e	2	$\mathbb{Q}(\sqrt{649})$	$2q + 15q^{2} + 40q^{3} + 181q^{4} + 949q^{6} - 600q^{7} + O(q^{8})$

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 $T$
25.8.a.a	$x - 62$
25.8.a.b	$x^{2} + 40x - 3324$
25.8.a.c	$x^{2} + 55x - 704$
25.8.a.d	$x^{2} - 464$
25.8.a.e	$x^{2} - 55x - 704$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(25))$	9
Old forms	6
Cusp forms $S_{8}(\Gamma_0(25))$	15
Eisenstein series $E_{8}(\Gamma_0(25))$	6
Modular forms $M_{8}(\Gamma_0(25))$	21

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(25)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.259 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, 5, 19] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 9 = 1 x 9 (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(25, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 25:8 1000

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