Atlas › Modular forms › Weight 4 › Level 162 › 162.4.a

Newspace 162.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(162))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
162.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} + 9q^{5} - 31q^{7} + O(q^{8})$	+
162.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} + 21q^{5} + 8q^{7} + O(q^{8})$	+
162.4.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} - 21q^{5} + 8q^{7} + O(q^{8})$	−
162.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} - 9q^{5} - 31q^{7} + O(q^{8})$	−
162.4.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 4q^{2} + 8q^{4} - 12q^{5} + 16q^{7} + O(q^{8})$	+
162.4.a.f	2	$\mathbb{Q}(\sqrt{105})$	$2q - 4q^{2} + 8q^{4} - 9q^{5} + 19q^{7} + O(q^{8})$	+
162.4.a.g	2	$\mathbb{Q}(\sqrt{105})$	$2q + 4q^{2} + 8q^{4} + 9q^{5} + 19q^{7} + O(q^{8})$	−
162.4.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{2} + 8q^{4} + 12q^{5} + 16q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
162.4.a.a	$x + 84$
162.4.a.b	$x - 45$
162.4.a.c	$x - 3$
162.4.a.d	$x + 102$
162.4.a.e	$x^{2} - 36x - 351$
162.4.a.f	$x^{2} - 48x - 3204$
162.4.a.g	$x^{2} - 66x + 144$
162.4.a.h	$x^{2} - 60x - 423$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(162))$	12
Old forms	57
Cusp forms $S_{4}(\Gamma_0(162))$	69
Eisenstein series $E_{4}(\Gamma_0(162))$	24
Modular forms $M_{4}(\Gamma_0(162))$	93

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(162)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(81))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.153 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, 6, 9, 18, 27, 54, 81, 162]: [1, 2, 2, 5, 4, 11, 10, 29, 28, 83] (certified at ell = 2147483179)
T = [(5, 1), (7, 3)]; 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(162, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 162:4 1000

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