Atlas › Modular forms › Weight 2 › Level 546 › 546.2.a

Newspace 546.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(546))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
546.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + + −	$+1$	546a1 rank 0
546.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − + −	$-1$	546c1 rank 1
546.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − + +	$+1$	546b1 rank 0
546.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − − −	$+1$	546d1 rank 0
546.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	− + + +	$+1$	546e1 rank 0
546.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	− − − +	$+1$	546f1 rank 0
546.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{7} + O(q^{8})$	− − − +	$+1$	546g1 rank 0
546.2.a.h	2	$\mathbb{Q}(\sqrt{57})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ + − +	$+1$
546.2.a.i	2	$\mathbb{Q}(\sqrt{41})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	− + − −	$+1$
546.2.a.j	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 3q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	− − + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{11} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{31} + 8T_{37}$ 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 $T$
546.2.a.a	$x - 131$
546.2.a.b	$x + 80$
546.2.a.c	$x - 59$
546.2.a.d	$x + 121$
546.2.a.e	$x + 27$
546.2.a.f	$x - 5$
546.2.a.g	$x + 16$
546.2.a.h	$x^{2} - 139x + 4816$
546.2.a.i	$x^{2} - 161x + 5240$
546.2.a.j	$x^{2} + 131x + 3776$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(546))$	13
Old forms	92
Cusp forms $S_{2}(\Gamma_0(546))$	105
Eisenstein series $E_{2}(\Gamma_0(546))$	15
Modular forms $M_{2}(\Gamma_0(546))$	120

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.599 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, 7, 13, 14, 21, 26, 39, 42, 78, 91, 182, 273, 546]: [0, 1, 1, 3, 1, 1, 4, 4, 5, 6, 12, 18, 10, 32, 40, 120] (certified at ell = 2147483629)
T = [(5, 1), (11, 2), (17, 3), (19, 4), (23, 5), (29, 6), (31, 7), (37, 8)]; h of degree 13 = 1 x 13 (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(546, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 546:2 1000

← 545.2.a · 547.2.a →