Atlas › Modular forms › Weight 6 › Level 13 › 13.6.a

Newspace 13.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(13))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
13.6.a.a	2	$\mathbb{Q}(\sqrt{17})$	$2q - 5q^{2} - 28q^{3} - 43q^{4} - 42q^{5} + 19q^{6} - 36q^{7} + O(q^{8})$	+	$-1$
13.6.a.b	3	degree 3	$3q + 7q^{2} + 8q^{3} + 121q^{4} + 56q^{5} - 199q^{6} - 60q^{7} + O(q^{8})$	−	$+1$

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 $T$
13.6.a.a	$x^{2} + 33x + 166$
13.6.a.b	$x^{3} - 15x^{2} - 66x + 144$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(13))$	5
Old forms	0
Cusp forms $S_{6}(\Gamma_0(13))$	5
Eisenstein series $E_{6}(\Gamma_0(13))$	2
Modular forms $M_{6}(\Gamma_0(13))$	7

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 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, 13]: [1, 7] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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(13, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 13:6 1000

← 12.6.a · 14.6.a →