Atlas › Modular forms › Weight 2 › Level 1000 › 1000.2.a

Newspace 1000.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(1000))$

Newforms

Label	Dim	Coefficient field	Trace form
1000.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} - q^{7} + O(q^{8})$
1000.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} + 4q^{7} + O(q^{8})$
1000.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{3} - 4q^{7} + O(q^{8})$
1000.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{3} + q^{7} + O(q^{8})$
1000.2.a.e	4	degree 4	$4q - 4q^{3} - 6q^{7} + O(q^{8})$
1000.2.a.f	4	degree 4	$4q - q^{3} - 9q^{7} + O(q^{8})$
1000.2.a.g	4	degree 4	$4q + q^{3} + 9q^{7} + O(q^{8})$
1000.2.a.h	4	degree 4	$4q + 4q^{3} + 6q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
1000.2.a.a	$x^{2} + 4x - 1$
1000.2.a.b	$x^{2} - 11x - 1$
1000.2.a.c	$x^{2} + 11x - 1$
1000.2.a.d	$x^{2} - 4x - 1$
1000.2.a.e	$x^{4} + 22x^{3} + 74x^{2} - 1042x - 5779$
1000.2.a.f	$x^{4} + 28x^{3} + 169x^{2} - 628x - 4999$
1000.2.a.g	$x^{4} - 28x^{3} + 169x^{2} + 628x - 4999$
1000.2.a.h	$x^{4} - 22x^{3} + 74x^{2} + 1042x - 5779$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(1000))$	24
Old forms	107
Cusp forms $S_{2}(\Gamma_0(1000))$	131
Eisenstein series $E_{2}(\Gamma_0(1000))$	39
Modular forms $M_{2}(\Gamma_0(1000))$	170

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(1000)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(100))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(125))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(200))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(250))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(500))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.498 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, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000]: [0, 1, 2, 1, 3, 3, 6, 3, 10, 9, 18, 13, 34, 39, 78, 154] (certified at ell = 2147482801)
T = [(3, 1), (7, 3)]; 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(1000, 2)

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

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

← 999.2.a · 1001.2.a →