Atlas › Modular forms › Weight 2 › Level 163 › 163.2.a

Newspace 163.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(163))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
163.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} - 4q^{5} + 2q^{7} + O(q^{8})$	+	$-1$	163a1 rank 1
163.2.a.b	5	degree 5	$5q - 5q^{2} - 5q^{3} + 9q^{4} - 9q^{5} - 3q^{6} - 6q^{7} + O(q^{8})$	+	$-1$
163.2.a.c	7	degree 7	$7q + 3q^{2} + q^{3} + 5q^{4} + 11q^{5} - 3q^{6} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
163.2.a.a	$x$
163.2.a.b	$x^{5} + 10x^{4} + 32x^{3} + 41x^{2} + 20x + 3$
163.2.a.c	$x^{7} - 4x^{6} - 10x^{5} + 55x^{4} - 14x^{3} - 149x^{2} + 144x - 8$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(163))$	13
Old forms	0
Cusp forms $S_{2}(\Gamma_0(163))$	13
Eisenstein series $E_{2}(\Gamma_0(163))$	1
Modular forms $M_{2}(\Gamma_0(163))$	14

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 163]: [0, 14] (certified at ell = 2147483179)
T = [(2, 1), (3, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 2 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(163, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 163:2 1000

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