Atlas › Modular forms › Weight 2 › Level 89 › 89.2.a

Newspace 89.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(89))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
89.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+	$-1$	89a1 rank 1
89.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	−	$+1$	89b1 rank 0
89.2.a.c	5	degree 5	$5q - q^{2} - 3q^{3} + 11q^{4} - q^{5} - 3q^{6} + 8q^{7} + O(q^{8})$	−	$+1$

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 $T$
89.2.a.a	$x + 2$
89.2.a.b	$x - 3$
89.2.a.c	$x^{5} + 4x^{4} - 8x^{3} - 6x^{2} + 10x - 2$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(89))$	7
Old forms	0
Cusp forms $S_{2}(\Gamma_0(89))$	7
Eisenstein series $E_{2}(\Gamma_0(89))$	1
Modular forms $M_{2}(\Gamma_0(89))$	8

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.163 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, 89]: [0, 8] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (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(89, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 89:2 1000

← 88.2.a · 90.2.a →