Atlas › Modular forms › Weight 2 › Level 564 › 564.2.a

Newspace 564.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(564))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
564.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - q^{7} + O(q^{8})$	+ −	564a1 rank 1
564.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} - q^{7} + O(q^{8})$	− +	564b1 rank 1
564.2.a.c	3	degree 3	$3q - 3q^{3} + 3q^{5} + 3q^{7} + O(q^{8})$	+ +
564.2.a.d	3	degree 3	$3q + 3q^{3} + 5q^{5} + 3q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
564.2.a.a	$x + 4$
564.2.a.b	$x + 6$
564.2.a.c	$x^{3} - 12x^{2} - 102x + 196$
564.2.a.d	$x^{3} - 14x^{2} + 10x + 296$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(564))$	8
Old forms	83
Cusp forms $S_{2}(\Gamma_0(564))$	91
Eisenstein series $E_{2}(\Gamma_0(564))$	11
Modular forms $M_{2}(\Gamma_0(564))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(564)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(141))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(188))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(282))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.349 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, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564]: [0, 1, 1, 2, 3, 5, 5, 14, 18, 27, 52, 102] (certified at ell = 2147483137)
T = [(5, 1), (7, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(564, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 564:2 1000

← 563.2.a · 565.2.a →