Atlas › Modular forms › Weight 2 › Level 512 › 512.2.a

Newspace 512.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(512))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
512.2.a.a	$x^{2} + 4x + 2$
512.2.a.b	$x^{2} + 12x + 34$
512.2.a.c	$x^{2} - 50$
512.2.a.d	$x^{2} - 98$
512.2.a.e	$x^{2} - 12x + 34$
512.2.a.f	$x^{2} - 4x + 2$
512.2.a.g	$x^{4} - 228x^{2} + 10404$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(512))$	16
Old forms	33
Cusp forms $S_{2}(\Gamma_0(512))$	49
Eisenstein series $E_{2}(\Gamma_0(512))$	31
Modular forms $M_{2}(\Gamma_0(512))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(512)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(128))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(256))^{\oplus 2}$

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, 2, 4, 8, 16, 32, 64, 128, 256, 512]: [0, 1, 2, 3, 4, 6, 10, 18, 34, 66] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 16 = 1 x 16 (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(512, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 512:2 1000

← 511.2.a · 513.2.a →