Atlas › Modular forms › Weight 2 › Level 561 › 561.2.a

Newspace 561.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(561))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
561.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - 2q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	561c1 rank 1
561.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} + O(q^{8})$	− − −	$+1$	561d1 rank 0
561.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - 2q^{5} - 3q^{7} + O(q^{8})$	+ − +	$+1$	561a1 rank 0
561.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 2q^{5} + q^{7} + O(q^{8})$	− − +	$-1$	561b1 rank 1
561.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 4q^{5} - 6q^{7} + O(q^{8})$	− + −	$-1$
561.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} - 2q^{3} + 5q^{4} + 4q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$
561.2.a.g	3	degree 3	$3q - 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$
561.2.a.h	3	degree 3	$3q - 3q^{3} + 2q^{4} - 4q^{5} - 7q^{7} + O(q^{8})$	+ − −	$-1$
561.2.a.i	3	degree 3	$3q + q^{2} - 3q^{3} + 3q^{4} + 4q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$
561.2.a.j	4	degree 4	$4q + q^{2} + 4q^{3} + 5q^{4} - 4q^{5} + q^{6} + 9q^{7} + O(q^{8})$	− − −	$+1$
561.2.a.k	6	degree 6	$6q - q^{2} + 6q^{3} + 11q^{4} + 2q^{5} - q^{6} + 7q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{13} + 7T_{19} + 9T_{23}$ 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 $T$
561.2.a.a	$x - 18$
561.2.a.b	$x + 21$
561.2.a.c	$x - 36$
561.2.a.d	$x + 132$
561.2.a.e	$x^{2} + 76x - 2974$
561.2.a.f	$x^{2} - 53x + 664$
561.2.a.g	$x^{3} + 36x^{2} - 2244x - 74990$
561.2.a.h	$x^{3} + 166x^{2} + 8658x + 142622$
561.2.a.i	$x^{3} - 81x^{2} + 1368x - 6534$
561.2.a.j	$x^{4} - 75x^{3} - 5942x^{2} + 439208x - 6205274$
561.2.a.k	$x^{6} - 129x^{5} + 3565x^{4} + 61739x^{3} - 1649240x^{2} - 16418782x + 34462734$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(561))$	27
Old forms	42
Cusp forms $S_{2}(\Gamma_0(561))$	69
Eisenstein series $E_{2}(\Gamma_0(561))$	7
Modular forms $M_{2}(\Gamma_0(561))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(561)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(187))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 3, 11, 17, 33, 51, 187, 561]: [0, 1, 2, 2, 6, 8, 20, 76] (certified at ell = 2147482801)
T = [(2, 1), (5, 3), (13, 5), (19, 7), (23, 9)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 7 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(561, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 561:2 1000

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