Atlas › Modular forms › Weight 10 › Level 13 › 13.10.a

Newspace 13.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(13))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
13.10.a.a	4	degree 4	$4q - 33q^{2} - 163q^{3} + 1429q^{4} + 471q^{5} - 4529q^{6} - 11241q^{7} + O(q^{8})$	+	$-1$
13.10.a.b	5	degree 5	$5q + 15q^{2} + 161q^{3} + 361q^{4} + 1803q^{5} + 5693q^{6} + 10099q^{7} + O(q^{8})$	−	$+1$

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 $T$
13.10.a.a	$x^{4} + 196x^{3} - 2391x^{2} - 923102x + 21397904$
13.10.a.b	$x^{5} - 176x^{4} - 71403x^{3} + 10141242x^{2} + 1035735264x - 113786339328$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(13))$	9
Old forms	0
Cusp forms $S_{10}(\Gamma_0(13))$	9
Eisenstein series $E_{10}(\Gamma_0(13))$	2
Modular forms $M_{10}(\Gamma_0(13))$	11

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 11] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 13:10 1000

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