Atlas › Modular forms › Weight 4 › Level 160 › 160.4.a

Newspace 160.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(160))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
160.4.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 5q^{5} + 6q^{7} + O(q^{8})$	+
160.4.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 5q^{5} - 6q^{7} + O(q^{8})$	+
160.4.a.c	2	$\mathbb{Q}(\sqrt{6})$	$2q - 8q^{3} + 10q^{5} - 8q^{7} + O(q^{8})$	−
160.4.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 10q^{5} + O(q^{8})$	+
160.4.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - 10q^{5} + O(q^{8})$	+
160.4.a.f	2	$\mathbb{Q}(\sqrt{10})$	$2q + 10q^{5} + O(q^{8})$	−
160.4.a.g	2	$\mathbb{Q}(\sqrt{6})$	$2q + 8q^{3} + 10q^{5} + 8q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
160.4.a.a	$x - 4$
160.4.a.b	$x + 4$
160.4.a.c	$x^{2} + 16x - 320$
160.4.a.d	$x^{2} - 720$
160.4.a.e	$x^{2} - 208$
160.4.a.f	$x^{2} - 640$
160.4.a.g	$x^{2} - 16x - 320$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(160))$	12
Old forms	52
Cusp forms $S_{4}(\Gamma_0(160))$	64
Eisenstein series $E_{4}(\Gamma_0(160))$	16
Modular forms $M_{4}(\Gamma_0(160))$	80

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(160)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 5}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(80))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160]: [1, 2, 3, 3, 5, 7, 8, 12, 14, 22, 40, 76] (certified at ell = 2147483497)
T = [(3, 1), (7, 1)]; h of degree 12 = 1 x 12 (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(160, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 160:4 1000

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