Atlas › Modular forms › Weight 2 › Level 558 › 558.2.a

Newspace 558.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(558))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
558.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} - 2q^{7} + O(q^{8})$	+ −	558d1 rank 1
558.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + O(q^{8})$	+ +	558a1 rank 1
558.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} + O(q^{8})$	+ +	558c1 rank 0
558.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	+ −	558b1 rank 0
558.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} - 4q^{7} + O(q^{8})$	− −	558f1 rank 1
558.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} - 2q^{7} + O(q^{8})$	− +	558g1 rank 1
558.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} + O(q^{8})$	− +	558e1 rank 0
558.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} + 2q^{7} + O(q^{8})$	− −	558h1 rank 0
558.2.a.i	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} + 2q^{4} - 3q^{5} + 2q^{7} + O(q^{8})$	+ +
558.2.a.j	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 2q^{4} + 4q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
558.2.a.a	$x + 26$
558.2.a.b	$x + 50$
558.2.a.c	$x - 70$
558.2.a.d	$x - 68$
558.2.a.e	$x + 22$
558.2.a.f	$x + 74$
558.2.a.g	$x - 36$
558.2.a.h	$x - 4$
558.2.a.i	$x^{2} - 38x - 1696$
558.2.a.j	$x^{2} - 28x - 5096$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(558))$	12
Old forms	77
Cusp forms $S_{2}(\Gamma_0(558))$	89
Eisenstein series $E_{2}(\Gamma_0(558))$	15
Modular forms $M_{2}(\Gamma_0(558))$	104

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(558)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(93))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(186))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(279))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.569 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, 3, 6, 9, 18, 31, 62, 93, 186, 279, 558]: [0, 1, 1, 3, 2, 5, 3, 10, 12, 36, 34, 100] (certified at ell = 2147482951)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 12 = 1 x 12 (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(558, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 558:2 1000

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