Atlas › Modular forms › Weight 2 › Level 520 › 520.2.a

Newspace 520.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(520))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
520.2.a.a	1	$\mathbb{Q}$	$q - q^{5} + O(q^{8})$	+ +	520a1 rank 1
520.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + O(q^{8})$	− −	520b1 rank 0
520.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	− +
520.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} - 2q^{5} + O(q^{8})$	+ −
520.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} - 2q^{5} + O(q^{8})$	+ −
520.2.a.f	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{5} + 4q^{7} + O(q^{8})$	− +
520.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} - 2q^{5} + O(q^{8})$	+ +

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{17} + 9T_{19} + 11T_{23}$ 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 $T$
520.2.a.a	$x + 26$
520.2.a.b	$x - 66$
520.2.a.c	$x^{2} + 160x + 4832$
520.2.a.d	$x^{2} + 188x + 8536$
520.2.a.e	$x^{2} - 68x + 1048$
520.2.a.f	$x^{2} - 72x - 3408$
520.2.a.g	$x^{2} - 92x - 764$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(520))$	12
Old forms	65
Cusp forms $S_{2}(\Gamma_0(520))$	77
Eisenstein series $E_{2}(\Gamma_0(520))$	15
Modular forms $M_{2}(\Gamma_0(520))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(520)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(104))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(130))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(260))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.622 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, 13, 20, 26, 40, 52, 65, 104, 130, 260, 520]: [0, 1, 2, 1, 3, 3, 1, 6, 5, 10, 10, 8, 18, 24, 48, 92] (certified at ell = 2147483629)
T = [(3, 1), (7, 3), (11, 5), (17, 7), (19, 9), (23, 11)]; h of degree 12 = 1 x 12 (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(520, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 520:2 1000

← 519.2.a · 521.2.a →