Atlas › Modular forms › Weight 2 › Level 573 › 573.2.a

Newspace 573.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(573))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
573.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	573c1 rank 1
573.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$	573a1 rank 0
573.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$	573b1 rank 0
573.2.a.d	5	degree 5	$5q - 3q^{2} + 5q^{3} + q^{4} - 5q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
573.2.a.e	6	degree 6	$6q - q^{2} - 6q^{3} + 3q^{4} + q^{5} + q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
573.2.a.f	7	degree 7	$7q + 3q^{2} + 7q^{3} + 7q^{4} - q^{5} + 3q^{6} + q^{7} + O(q^{8})$	− +	$+1$
573.2.a.g	10	degree 10	$10q + 3q^{2} - 10q^{3} + 15q^{4} + q^{5} - 3q^{6} + 11q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
573.2.a.a	$x + 4$
573.2.a.b	$x - 1$
573.2.a.c	$x - 4$
573.2.a.d	$x^{5} + 8x^{4} + 12x^{3} - 27x^{2} - 36x + 37$
573.2.a.e	$x^{6} - 10x^{4} + x^{3} + 16x^{2} - 5x - 2$
573.2.a.f	$x^{7} - 2x^{6} - 23x^{5} + 57x^{4} + 26x^{3} - 30x^{2} + 2x + 1$
573.2.a.g	$x^{10} - 4x^{9} - 48x^{8} + 177x^{7} + 801x^{6} - 2851x^{5} - 5344x^{4} + 19063x^{3} + 11040x^{2} - 40937x - 2026$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(573))$	31
Old forms	32
Cusp forms $S_{2}(\Gamma_0(573))$	63
Eisenstein series $E_{2}(\Gamma_0(573))$	3
Modular forms $M_{2}(\Gamma_0(573))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(573)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(191))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.216 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, 191, 573]: [0, 1, 17, 66] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 31 = 1 x 31 (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(573, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 573:2 1000

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