Atlas › Modular forms › Weight 2 › Level 162 › 162.2.a

Newspace 162.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(162))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
162.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 3q^{5} - 4q^{7} + O(q^{8})$	+	162a1 rank 1
162.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{7} + O(q^{8})$	+	162c1 rank 0
162.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{7} + O(q^{8})$	−	162b1 rank 0
162.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	−	162d1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
162.2.a.a	$x + 22$
162.2.a.b	$x - 35$
162.2.a.c	$x - 5$
162.2.a.d	$x + 16$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(162))$	4
Old forms	12
Cusp forms $S_{2}(\Gamma_0(162))$	16
Eisenstein series $E_{2}(\Gamma_0(162))$	23
Modular forms $M_{2}(\Gamma_0(162))$	39

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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]: [0, 1, 1, 3, 2, 5, 4, 11, 10, 29] (certified at ell = 2147483179)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 4 = 1 x 4 (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(162, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 162:2 1000

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