Atlas › Modular forms › Weight 2 › Level 594 › 594.2.a

Newspace 594.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(594))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
594.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 3q^{5} - 4q^{7} + O(q^{8})$	+ +	594c1 rank 0
594.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} - q^{7} + O(q^{8})$	+ −	594d1 rank 1
594.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} + q^{7} + O(q^{8})$	+ +	594a1 rank 1
594.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + 4q^{7} + O(q^{8})$	+ +	594b1 rank 0
594.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} + 4q^{7} + O(q^{8})$	− −	594f1 rank 0
594.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} - q^{7} + O(q^{8})$	− +	594e1 rank 0
594.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + q^{7} + O(q^{8})$	− −	594g1 rank 0
594.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	− −	594h1 rank 0
594.2.a.i	2	$\mathbb{Q}(\sqrt{10})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	+ −
594.2.a.j	2	$\mathbb{Q}(\sqrt{10})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
594.2.a.a	$x + 92$
594.2.a.b	$x + 215$
594.2.a.c	$x + 129$
594.2.a.d	$x - 204$
594.2.a.e	$x - 6$
594.2.a.f	$x - 3$
594.2.a.g	$x - 33$
594.2.a.h	$x + 74$
594.2.a.i	$x^{2} - 476x + 55204$
594.2.a.j	$x^{2} + 104x + 744$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(594))$	12
Old forms	85
Cusp forms $S_{2}(\Gamma_0(594))$	97
Eisenstein series $E_{2}(\Gamma_0(594))$	23
Modular forms $M_{2}(\Gamma_0(594))$	120

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(594)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(66))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(99))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(198))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(297))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.851 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, 11, 18, 22, 27, 33, 54, 66, 99, 198, 297, 594]: [0, 1, 1, 3, 2, 2, 5, 5, 4, 6, 11, 16, 14, 40, 38, 112] (certified at ell = 2147482801)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; 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(594, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 594:2 1000

← 593.2.a · 595.2.a →