Atlas › Modular forms › Weight 6 › Level 25 › 25.6.a

Newspace 25.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(25))$

Newforms

Label	Dim	Coefficient field	Trace form
25.6.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{3} - 28q^{4} - 8q^{6} - 192q^{7} + O(q^{8})$
25.6.a.b	2	$\mathbb{Q}(\sqrt{241})$	$2q - 5q^{2} - 20q^{3} + 69q^{4} - 191q^{6} - 200q^{7} + O(q^{8})$
25.6.a.c	2	$\mathbb{Q}(\sqrt{11})$	$2q + 24q^{4} + 264q^{6} + O(q^{8})$
25.6.a.d	2	$\mathbb{Q}(\sqrt{241})$	$2q + 5q^{2} + 20q^{3} + 69q^{4} - 191q^{6} + 200q^{7} + O(q^{8})$

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 $T$
25.6.a.a	$x - 2$
25.6.a.b	$x^{2} + 25x + 96$
25.6.a.c	$x^{2} - 704$
25.6.a.d	$x^{2} - 25x + 96$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(25))$	7
Old forms	2
Cusp forms $S_{6}(\Gamma_0(25))$	9
Eisenstein series $E_{6}(\Gamma_0(25))$	6
Modular forms $M_{6}(\Gamma_0(25))$	15

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(25)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.217 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, 13] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (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(25, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 25:6 1000

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