Atlas › Modular forms › Weight 2 › Level 568 › 568.2.a

Newspace 568.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(568))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
568.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + 5q^{7} + O(q^{8})$	−	568a1 rank 0
568.2.a.b	4	degree 4	$4q - 4q^{3} - q^{5} - 5q^{7} + O(q^{8})$	+
568.2.a.c	4	degree 4	$4q - 2q^{3} - 7q^{5} + q^{7} + O(q^{8})$	−
568.2.a.d	4	degree 4	$4q + 5q^{3} - q^{5} + O(q^{8})$	−
568.2.a.e	5	degree 5	$5q + 2q^{3} + 7q^{5} - q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
568.2.a.a	$x - 5$
568.2.a.b	$x^{4} + 7x^{3} - 35x^{2} + 22x + 1$
568.2.a.c	$x^{4} + 23x^{3} + 157x^{2} + 250x - 307$
568.2.a.d	$x^{4} - 2x^{3} - 139x^{2} - 103x + 1550$
568.2.a.e	$x^{5} - 23x^{4} + 87x^{3} + 1236x^{2} - 10403x + 21218$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(568))$	18
Old forms	51
Cusp forms $S_{2}(\Gamma_0(568))$	69
Eisenstein series $E_{2}(\Gamma_0(568))$	7
Modular forms $M_{2}(\Gamma_0(568))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(568)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(142))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(284))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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, 8, 71, 142, 284, 568]: [0, 1, 2, 3, 7, 20, 39, 76] (certified at ell = 2147483171)
T = [(3, 1), (5, 3)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 3 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(568, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 568:2 1000

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