Atlas › Modular forms › Weight 2 › Level 586 › 586.2.a

Newspace 586.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(586))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
586.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 3q^{5} - 2q^{6} + O(q^{8})$	+ −	$+1$	586a1 rank 0
586.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	− −	$-1$	586c1 rank 1
586.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$	586b1 rank 1
586.2.a.d	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + 2q^{5} + 3q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
586.2.a.e	3	degree 3	$3q - 3q^{2} + 3q^{4} - 6q^{5} + O(q^{8})$	+ +	$-1$
586.2.a.f	5	degree 5	$5q - 5q^{2} + 5q^{4} - 4q^{5} - 3q^{7} + O(q^{8})$	+ −	$+1$
586.2.a.g	10	degree 10	$10q + 10q^{2} - q^{3} + 10q^{4} + 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$

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 $T$
586.2.a.a	$x - 5$
586.2.a.b	$x + 3$
586.2.a.c	$x + 1$
586.2.a.d	$x^{3} + x^{2} - 13x + 5$
586.2.a.e	$x^{3} + 6x^{2} + 5x + 1$
586.2.a.f	$x^{5} + 4x^{4} - 12x^{3} - 21x^{2} + 25x + 5$
586.2.a.g	$x^{10} - 2x^{9} - 60x^{8} + 208x^{7} + 708x^{6} - 4170x^{5} + 4959x^{4} + 3110x^{3} - 9647x^{2} + 6292x - 1331$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(586))$	24
Old forms	48
Cusp forms $S_{2}(\Gamma_0(586))$	72
Eisenstein series $E_{2}(\Gamma_0(586))$	3
Modular forms $M_{2}(\Gamma_0(586))$	75

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(586)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(293))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.212 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, 293, 586]: [0, 1, 25, 75] (certified at ell = 2147478961)
T = [(3, 1), (5, 1)]; h of degree 24 = 1 x 24 (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(586, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 586:2 1000

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