Atlas › Modular forms › Weight 2 › Level 560 › 560.2.a

Newspace 560.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(560))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
560.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - q^{5} + q^{7} + O(q^{8})$	+ −	560e1 rank 1
560.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - q^{7} + O(q^{8})$	+ +	560c1 rank 0
560.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - q^{7} + O(q^{8})$	− +	560f1 rank 1
560.2.a.d	1	$\mathbb{Q}$	$q - q^{5} + q^{7} + O(q^{8})$	+ −	560d1 rank 1
560.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - q^{5} + q^{7} + O(q^{8})$	+ −	560a1 rank 0
560.2.a.f	1	$\mathbb{Q}$	$q + 3q^{3} + q^{5} - q^{7} + O(q^{8})$	− +	560b1 rank 0
560.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	− +
560.2.a.h	2	$\mathbb{Q}(\sqrt{33})$	$2q + q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	+ −
560.2.a.i	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{11} + 3T_{13} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{17} + 8T_{31}$ 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 $T$
560.2.a.a	$x + 85$
560.2.a.b	$x - 113$
560.2.a.c	$x + 127$
560.2.a.d	$x + 40$
560.2.a.e	$x - 35$
560.2.a.f	$x + 23$
560.2.a.g	$x^{2} - 177x + 5176$
560.2.a.h	$x^{2} - 123x + 3576$
560.2.a.i	$x^{2} - 19x - 1784$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(560))$	12
Old forms	73
Cusp forms $S_{2}(\Gamma_0(560))$	85
Eisenstein series $E_{2}(\Gamma_0(560))$	23
Modular forms $M_{2}(\Gamma_0(560))$	108

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.684 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, 7, 8, 10, 14, 16, 20, 28, 35, 40, 56, 70, 80, 112, 140, 280, 560]: [0, 1, 2, 1, 1, 3, 3, 4, 4, 6, 7, 6, 10, 12, 16, 16, 20, 30, 56, 104] (certified at ell = 2147483629)
T = [(3, 1), (11, 2), (13, 3), (19, 4), (23, 5), (29, 6), (17, 7), (31, 8)]; 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(560, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 560:2 1000

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