Atlas › Modular forms › Weight 4 › Level 168 › 168.4.a

Newspace 168.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(168))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
168.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 10q^{5} + 7q^{7} + O(q^{8})$	+ −
168.4.a.b	1	$\mathbb{Q}$	$q - 3q^{3} - 2q^{5} + 7q^{7} + O(q^{8})$	+ −
168.4.a.c	1	$\mathbb{Q}$	$q - 3q^{3} + 4q^{5} - 7q^{7} + O(q^{8})$	+ +
168.4.a.d	1	$\mathbb{Q}$	$q + 3q^{3} - 16q^{5} + 7q^{7} + O(q^{8})$	− −
168.4.a.e	1	$\mathbb{Q}$	$q + 3q^{3} - 10q^{5} - 7q^{7} + O(q^{8})$	− +
168.4.a.f	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{5} - 7q^{7} + O(q^{8})$	− +
168.4.a.g	2	$\mathbb{Q}(\sqrt{337})$	$2q - 6q^{3} - 6q^{5} - 14q^{7} + O(q^{8})$	+ +
168.4.a.h	2	$\mathbb{Q}(\sqrt{177})$	$2q + 6q^{3} + 14q^{5} + 14q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19} + 11T_{23}$ 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 $T$
168.4.a.a	$x - 3346$
168.4.a.b	$x + 1438$
168.4.a.c	$x + 2254$
168.4.a.d	$x + 2990$
168.4.a.e	$x + 534$
168.4.a.f	$x - 1658$
168.4.a.g	$x^{2} - 1700x - 657852$
168.4.a.h	$x^{2} - 1612x + 578836$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(168))$	10
Old forms	78
Cusp forms $S_{4}(\Gamma_0(168))$	88
Eisenstein series $E_{4}(\Gamma_0(168))$	16
Modular forms $M_{4}(\Gamma_0(168))$	104

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.39 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, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168]: [1, 2, 2, 3, 5, 3, 5, 9, 8, 10, 16, 15, 28, 28, 54, 104] (certified at ell = 2147483647)
T = [(5, 1), (11, 3), (13, 5), (17, 7), (19, 9), (23, 11)]; h of degree 10 = 1 x 10 (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(168, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 168:4 1000

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