Atlas › Modular forms › Weight 4 › Level 159 › 159.4.a

Newspace 159.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(159))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
159.4.a.a	1	$\mathbb{Q}$	$q - 5q^{2} - 3q^{3} + 17q^{4} - 21q^{5} + 15q^{6} - 16q^{7} + O(q^{8})$	+ +	$+1$
159.4.a.b	1	$\mathbb{Q}$	$q - 5q^{2} + 3q^{3} + 17q^{4} + 3q^{5} - 15q^{6} - 16q^{7} + O(q^{8})$	− +	$-1$
159.4.a.c	3	degree 3	$3q - q^{2} + 9q^{3} - 11q^{4} - 17q^{5} - 3q^{6} - 8q^{7} + O(q^{8})$	− +	$-1$
159.4.a.d	4	degree 4	$4q - 12q^{3} + 6q^{4} + 2q^{5} - 24q^{7} + O(q^{8})$	+ −	$-1$
159.4.a.e	8	degree 8	$8q + 5q^{2} - 24q^{3} + 29q^{4} + 23q^{5} - 15q^{6} + 20q^{7} + O(q^{8})$	+ +	$+1$
159.4.a.f	9	degree 9	$9q + 6q^{2} + 27q^{3} + 46q^{4} + 26q^{5} + 18q^{6} + 4q^{7} + O(q^{8})$	− −	$+1$

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 $T$
159.4.a.a	$x + 26$
159.4.a.b	$x + 2$
159.4.a.c	$x^{3} + 18x^{2} - 40x - 1291$
159.4.a.d	$x^{4} - 2x^{3} - 82x^{2} - 267x - 134$
159.4.a.e	$x^{8} - 28x^{7} - 267x^{6} + 11155x^{5} - 6153x^{4} - 1082810x^{3} + 2953046x^{2} + 8919135x - 25355655$
159.4.a.f	$x^{9} - 32x^{8} - 269x^{7} + 16443x^{6} - 106461x^{5} - 806056x^{4} + 7508174x^{3} - 2349347x^{2} - 67101639x + 102573198$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(159))$	26
Old forms	26
Cusp forms $S_{4}(\Gamma_0(159))$	52
Eisenstein series $E_{4}(\Gamma_0(159))$	4
Modular forms $M_{4}(\Gamma_0(159))$	56

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(159)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.124 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]: [1, 2, 15, 56] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 5 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 159:4 1000

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