Atlas › Modular forms › Weight 2 › Level 500 › 500.2.a

Newspace 500.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(500))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
500.2.a.a	$x^{2} + 5x + 5$
500.2.a.b	$x^{2} - 5x + 5$
500.2.a.c	$x^{4} - 60x^{2} + 775$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(500))$	8
Old forms	53
Cusp forms $S_{2}(\Gamma_0(500))$	61
Eisenstein series $E_{2}(\Gamma_0(500))$	29
Modular forms $M_{2}(\Gamma_0(500))$	90

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.227 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, 10, 20, 25, 50, 100, 125, 250, 500]: [0, 1, 2, 1, 3, 6, 3, 9, 18, 13, 39, 78] (certified at ell = 2147482801)
T = [(3, 1), (7, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 1 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(500, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 500:2 1000

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