Atlas › Modular forms › Weight 2 › Level 625 › 625.2.a

Newspace 625.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(625))$

Newforms

Label	Dim	Coefficient field	Trace form
625.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 3q^{3} - q^{4} + 4q^{6} - q^{7} + O(q^{8})$
625.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - q^{6} - q^{7} + O(q^{8})$
625.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} - q^{6} + q^{7} + O(q^{8})$
625.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 3q^{3} - q^{4} + 4q^{6} + q^{7} + O(q^{8})$
625.2.a.e	8	degree 8	$8q - 5q^{2} - 5q^{3} + 11q^{4} - 4q^{6} - 10q^{7} + O(q^{8})$
625.2.a.f	8	degree 8	$8q + 6q^{4} + 6q^{6} + O(q^{8})$
625.2.a.g	8	degree 8	$8q + 5q^{2} + 5q^{3} + 11q^{4} - 4q^{6} + 10q^{7} + O(q^{8})$

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 $T$
625.2.a.a	$x^{2} + 4x - 1$
625.2.a.b	$x^{2} - x - 1$
625.2.a.c	$x^{2} + x - 1$
625.2.a.d	$x^{2} - 4x - 1$
625.2.a.e	$x^{8} + 10x^{7} + 24x^{6} - 35x^{5} - 154x^{4} - 25x^{3} + 124x^{2} + 45x + 1$
625.2.a.f	$x^{8} - 31x^{6} + 276x^{4} - 851x^{2} + 841$
625.2.a.g	$x^{8} - 10x^{7} + 24x^{6} + 35x^{5} - 154x^{4} + 25x^{3} + 124x^{2} - 45x + 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(625))$	32
Old forms	16
Cusp forms $S_{2}(\Gamma_0(625))$	48
Eisenstein series $E_{2}(\Gamma_0(625))$	29
Modular forms $M_{2}(\Gamma_0(625))$	77

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(625)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(125))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 625]: [0, 1, 3, 13, 63] (certified at ell = 2147482501)
T = [(2, 1), (3, 1)]; h of degree 32 = 1 x 32 (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(625, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 625:2 1000

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