Atlas › Modular forms › Weight 2 › Level 203 › 203.2.a

Newspace 203.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(203))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
203.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− +	$+1$	203a1 rank 0
203.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + q^{7} + O(q^{8})$	− −	$-1$	203b1 rank 1
203.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} + 2q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− +	$+1$	203c1 rank 0
203.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - q^{3} - 2q^{4} + 3q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$
203.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{2} + 2q^{3} + 4q^{4} + 4q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$
203.2.a.f	3	degree 3	$3q - q^{2} - 3q^{3} + q^{4} - 5q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
203.2.a.g	5	degree 5	$5q + 2q^{2} - 2q^{3} + 10q^{4} + 5q^{5} - 9q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
203.2.a.a	$x + 3$
203.2.a.b	$x + 2$
203.2.a.c	$x - 3$
203.2.a.d	$x^{2} + 3x - 2$
203.2.a.e	$x^{2} - 6x + 7$
203.2.a.f	$x^{3} + 4x^{2} - 4$
203.2.a.g	$x^{5} - 13x^{3} + 40x - 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(203))$	15
Old forms	4
Cusp forms $S_{2}(\Gamma_0(203))$	19
Eisenstein series $E_{2}(\Gamma_0(203))$	3
Modular forms $M_{2}(\Gamma_0(203))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(203)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 7, 29, 203]: [0, 1, 3, 22] (certified at ell = 2147483353)
T = [(2, 1), (3, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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(203, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 203:2 1000

← 202.2.a · 204.2.a →