Atlas › Modular forms › Weight 2 › Level 585 › 585.2.a

Newspace 585.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(585))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
585.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - q^{5} - 3q^{7} + O(q^{8})$	+ −	585g1 rank 1
585.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - q^{5} + 3q^{7} + O(q^{8})$	+ +	585e1 rank 0
585.2.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} + q^{5} - q^{7} + O(q^{8})$	− +	585i1 rank 1
585.2.a.d	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{5} + 2q^{7} + O(q^{8})$	+ +	585a1 rank 1
585.2.a.e	1	$\mathbb{Q}$	$q - 2q^{4} - q^{5} - q^{7} + O(q^{8})$	+ −	585b1 rank 0
585.2.a.f	1	$\mathbb{Q}$	$q - 2q^{4} + q^{5} - q^{7} + O(q^{8})$	− −	585d1 rank 1
585.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} + O(q^{8})$	+ −	585f1 rank 1
585.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - 4q^{7} + O(q^{8})$	− +	585h1 rank 1
585.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} + 2q^{7} + O(q^{8})$	− +	585c1 rank 0
585.2.a.j	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 5q^{4} - 2q^{5} - 5q^{7} + O(q^{8})$	+ +
585.2.a.k	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− −
585.2.a.l	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 5q^{4} + 2q^{5} - 5q^{7} + O(q^{8})$	− +
585.2.a.m	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	+ +
585.2.a.n	3	degree 3	$3q + 8q^{4} + 3q^{5} + q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
585.2.a.a	$x + 83$
585.2.a.b	$x + 15$
585.2.a.c	$x + 37$
585.2.a.d	$x + 103$
585.2.a.e	$x - 111$
585.2.a.f	$x + 123$
585.2.a.g	$x + 137$
585.2.a.h	$x + 109$
585.2.a.i	$x - 11$
585.2.a.j	$x^{2} - 64x + 599$
585.2.a.k	$x^{2} - 144x + 4677$
585.2.a.l	$x^{2} - 88x - 937$
585.2.a.m	$x^{2} - 130x + 3887$
585.2.a.n	$x^{3} + 3x^{2} - 4345x + 104197$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(585))$	20
Old forms	57
Cusp forms $S_{2}(\Gamma_0(585))$	77
Eisenstein series $E_{2}(\Gamma_0(585))$	15
Modular forms $M_{2}(\Gamma_0(585))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(585)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(117))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(195))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.586 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, 5, 9, 13, 15, 39, 45, 65, 117, 195, 585]: [0, 1, 1, 2, 1, 4, 6, 8, 8, 16, 32, 88] (certified at ell = 2147483629)
T = [(2, 1), (7, 2), (11, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 20 = 1 x 20 (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(585, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 585:2 1000

← 584.2.a · 586.2.a →