Atlas › Modular forms › Weight 4 › Level 13 › 13.4.a

Newspace 13.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(13))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
13.4.a.a	1	$\mathbb{Q}$	$q - 5q^{2} - 7q^{3} + 17q^{4} - 7q^{5} + 35q^{6} - 13q^{7} + O(q^{8})$	−	$-1$
13.4.a.b	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 5q^{3} - 7q^{4} - 3q^{5} - 23q^{6} - 9q^{7} + O(q^{8})$	+	$+1$

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 $T$
13.4.a.a	$x + 12$
13.4.a.b	$x^{2} - 6x - 8$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(13))$	3
Old forms	0
Cusp forms $S_{4}(\Gamma_0(13))$	3
Eisenstein series $E_{4}(\Gamma_0(13))$	2
Modular forms $M_{4}(\Gamma_0(13))$	5

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.094 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, 5] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 13:4 1000

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