Atlas › Modular forms › Weight 2 › Level 159 › 159.2.a

Newspace 159.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(159))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
159.2.a.a	4	degree 4	$4q + 3q^{2} + 4q^{3} + 3q^{4} + 2q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	− +	$+1$
159.2.a.b	5	degree 5	$5q - 5q^{3} + 10q^{4} + 4q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
159.2.a.a	$x^{4} - 5x^{3} + 4x^{2} + 8x - 9$
159.2.a.b	$x^{5} - 37x^{3} + 8x^{2} + 195x - 99$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(159))$	9
Old forms	8
Cusp forms $S_{2}(\Gamma_0(159))$	17
Eisenstein series $E_{2}(\Gamma_0(159))$	3
Modular forms $M_{2}(\Gamma_0(159))$	20

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(159)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.085 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, 3, 53, 159]: [0, 1, 5, 20] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 9 = 1 x 9 (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(159, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 159:2 1000

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