Atlas › Modular forms › Weight 8 › Level 13 › 13.8.a

Newspace 13.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(13))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
13.8.a.a	1	$\mathbb{Q}$	$q + 10q^{2} - 73q^{3} - 28q^{4} - 295q^{5} - 730q^{6} + 1373q^{7} + O(q^{8})$	−	$-1$
13.8.a.b	2	$\mathbb{Q}(\sqrt{337})$	$2q - 19q^{2} + 45q^{3} + 93q^{4} - 353q^{5} - 933q^{6} - 2009q^{7} + O(q^{8})$	−	$-1$
13.8.a.c	4	degree 4	$4q + 15q^{2} + 80q^{3} + 253q^{4} + 258q^{5} + 1579q^{6} + 1692q^{7} + O(q^{8})$	+	$+1$

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 $T$
13.8.a.a	$x + 63$
13.8.a.b	$x^{2} - 26x - 168$
13.8.a.c	$x^{4} - 95x^{3} - 3570x^{2} + 411472x - 3668752$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(13))$	7
Old forms	0
Cusp forms $S_{8}(\Gamma_0(13))$	7
Eisenstein series $E_{8}(\Gamma_0(13))$	2
Modular forms $M_{8}(\Gamma_0(13))$	9

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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, 9] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (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(13, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 13:8 1000

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