Atlas › Modular forms › Weight 2 › Level 566 › 566.2.a

Newspace 566.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(566))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
566.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{7} + O(q^{8})$	+ +	$-1$	566a1 rank 1
566.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$	566b1 rank 0
566.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} + q^{3} + 2q^{4} - q^{5} - q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
566.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
566.2.a.e	4	degree 4	$4q - 4q^{2} + q^{3} + 4q^{4} - 2q^{5} - q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$
566.2.a.f	5	degree 5	$5q - 5q^{2} - 3q^{3} + 5q^{4} + 6q^{5} + 3q^{6} + 11q^{7} + O(q^{8})$	+ −	$+1$
566.2.a.g	9	degree 9	$9q + 9q^{2} + 3q^{3} + 9q^{4} + 3q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

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 $T$
566.2.a.a	$x$
566.2.a.b	$x + 5$
566.2.a.c	$x^{2} + 2x - 12$
566.2.a.d	$x^{2} + 6x + 4$
566.2.a.e	$x^{4} + 5x^{3} - 56x^{2} - 294x - 343$
566.2.a.f	$x^{5} - 15x^{4} - 118x^{3} + 2892x^{2} - 12560x + 4236$
566.2.a.g	$x^{9} - 3x^{8} - 286x^{7} + 1854x^{6} + 20897x^{5} - 211878x^{4} - 69568x^{3} + 6405804x^{2} - 23208940x + 24293744$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(566))$	24
Old forms	46
Cusp forms $S_{2}(\Gamma_0(566))$	70
Eisenstein series $E_{2}(\Gamma_0(566))$	3
Modular forms $M_{2}(\Gamma_0(566))$	73

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(566)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(283))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 283, 566]: [0, 1, 24, 73] (certified at ell = 2147480197)
T = [(3, 1), (5, 3)]; h of degree 24 = 1 x 24 (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(566, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 566:2 1000

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