Atlas › Modular forms › Weight 2 › Level 459 › 459.2.a

Newspace 459.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(459))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
459.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - 4q^{5} + q^{7} + O(q^{8})$	+	459b1 rank 1
459.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+	459g1 rank 0
459.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - 2q^{7} + O(q^{8})$	−	459h1 rank 1
459.2.a.d	1	$\mathbb{Q}$	$q - 2q^{4} - 3q^{5} + 2q^{7} + O(q^{8})$	+	459f1 rank 0
459.2.a.e	1	$\mathbb{Q}$	$q - 2q^{4} + 3q^{5} + 2q^{7} + O(q^{8})$	−	459c1 rank 0
459.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} - 2q^{7} + O(q^{8})$	+	459a1 rank 1
459.2.a.g	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	−	459d1 rank 0
459.2.a.h	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 4q^{5} + q^{7} + O(q^{8})$	−	459e1 rank 0
459.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{4} + 3q^{5} - 3q^{7} + O(q^{8})$	−
459.2.a.j	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 3q^{4} - 5q^{5} + 3q^{7} + O(q^{8})$	−
459.2.a.k	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - 3q^{5} - 3q^{7} + O(q^{8})$	+
459.2.a.l	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} + 3q^{4} + 5q^{5} + 3q^{7} + O(q^{8})$	+
459.2.a.m	3	degree 3	$3q - q^{2} + 9q^{4} - 3q^{5} - 4q^{7} + O(q^{8})$	−
459.2.a.n	3	degree 3	$3q + q^{2} + 9q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
459.2.a.a	$x + 96$
459.2.a.b	$x - 78$
459.2.a.c	$x + 101$
459.2.a.d	$x - 76$
459.2.a.e	$x - 16$
459.2.a.f	$x - 27$
459.2.a.g	$x + 32$
459.2.a.h	$x - 28$
459.2.a.i	$x^{2} - 191x + 8339$
459.2.a.j	$x^{2} + 109x + 2239$
459.2.a.k	$x^{2} + 117x + 2511$
459.2.a.l	$x^{2} - 43x + 459$
459.2.a.m	$x^{3} - 3x^{2} - 4949x + 108373$
459.2.a.n	$x^{3} - 43x^{2} - 249x + 15669$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(459))$	22
Old forms	27
Cusp forms $S_{2}(\Gamma_0(459))$	49
Eisenstein series $E_{2}(\Gamma_0(459))$	11
Modular forms $M_{2}(\Gamma_0(459))$	60

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(459)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(153))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 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, 3, 9, 17, 27, 51, 153, 459]: [0, 1, 2, 2, 4, 8, 20, 56] (certified at ell = 2147482801)
T = [(2, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (23, 7), (29, 8)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 6 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(459, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 459:2 1000

← 458.2.a · 460.2.a →