Atlas › Modular forms › Weight 2 › Level 537 › 537.2.a

Newspace 537.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(537))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
537.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$	537e1 rank 0
537.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 3q^{5} + 2q^{7} + O(q^{8})$	− +	$+1$	537c1 rank 0
537.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{5} + O(q^{8})$	− +	$+1$	537b1 rank 0
537.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	537a1 rank 0
537.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + 4q^{5} + q^{6} + q^{7} + O(q^{8})$	− +	$+1$	537d1 rank 0
537.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 2q^{3} + 5q^{4} + q^{5} + q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
537.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{2} + 2q^{3} + 4q^{4} + 2q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
537.2.a.h	6	degree 6	$6q - 4q^{2} + 6q^{3} + 4q^{4} - 10q^{5} - 4q^{6} - 10q^{7} + O(q^{8})$	− −	$-1$
537.2.a.i	6	degree 6	$6q - 2q^{2} - 6q^{3} + 4q^{4} + 4q^{5} + 2q^{6} - 12q^{7} + O(q^{8})$	+ +	$-1$
537.2.a.j	8	degree 8	$8q + 2q^{2} - 8q^{3} + 14q^{4} - 6q^{5} - 2q^{6} + 15q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
537.2.a.a	$x - 5$
537.2.a.b	$x - 43$
537.2.a.c	$x - 3$
537.2.a.d	$x - 38$
537.2.a.e	$x - 32$
537.2.a.f	$x^{2} - 47x + 344$
537.2.a.g	$x^{2} - 2x - 31$
537.2.a.h	$x^{6} + 112x^{5} + 3547x^{4} - 8472x^{3} - 2125369x^{2} - 33175460x - 154061275$
537.2.a.i	$x^{6} + 78x^{5} + 699x^{4} - 55810x^{3} - 759047x^{2} + 9730512x + 72862411$
537.2.a.j	$x^{8} - 31x^{7} - 2231x^{6} + 56853x^{5} + 1657817x^{4} - 31911549x^{3} - 414703161x^{2} + 5648282331x + 19247837866$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(537))$	29
Old forms	30
Cusp forms $S_{2}(\Gamma_0(537))$	59
Eisenstein series $E_{2}(\Gamma_0(537))$	3
Modular forms $M_{2}(\Gamma_0(537))$	62

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(537)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(179))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.316 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, 179, 537]: [0, 1, 16, 62] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 7 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(537, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 537:2 1000

← 536.2.a · 538.2.a →