Atlas › Modular forms › Weight 2 › Level 572 › 572.2.a

Newspace 572.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(572))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
572.2.a.a	1	$\mathbb{Q}$	$q + q^{3} + 3q^{5} + 2q^{7} + O(q^{8})$	− −	572a1 rank 0
572.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q - 3q^{3} - 3q^{7} + O(q^{8})$	+ −
572.2.a.c	2	$\mathbb{Q}(\sqrt{21})$	$2q + q^{3} - 4q^{5} + 3q^{7} + O(q^{8})$	− −
572.2.a.d	2	$\mathbb{Q}(\sqrt{21})$	$2q + q^{3} + 4q^{5} + 5q^{7} + O(q^{8})$	+ +
572.2.a.e	3	degree 3	$3q - 2q^{3} - q^{5} - 7q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
572.2.a.a	$x - 4$
572.2.a.b	$x^{2} + 3x - 1$
572.2.a.c	$x^{2} + 3x - 3$
572.2.a.d	$x^{2} - 5x + 1$
572.2.a.e	$x^{3} + 3x^{2} - x - 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(572))$	10
Old forms	69
Cusp forms $S_{2}(\Gamma_0(572))$	79
Eisenstein series $E_{2}(\Gamma_0(572))$	11
Modular forms $M_{2}(\Gamma_0(572))$	90

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(572)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(143))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(286))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 11, 13, 22, 26, 44, 52, 143, 286, 572]: [0, 1, 2, 2, 1, 5, 5, 9, 10, 16, 46, 90] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (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(572, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 572:2 1000

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