Atlas › Modular forms › Weight 2 › Level 109 › 109.2.a

Newspace 109.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(109))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
109.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 3q^{5} + 2q^{7} + O(q^{8})$	−	$+1$	109a1 rank 0
109.2.a.b	3	degree 3	$3q - 2q^{2} - 4q^{3} - 6q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+	$-1$
109.2.a.c	4	degree 4	$4q - q^{2} + 4q^{3} + 3q^{4} + q^{5} - 3q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
109.2.a.a	$x - 1$
109.2.a.b	$x^{3} + 14x^{2} + 56x + 56$
109.2.a.c	$x^{4} - 11x^{3} - 26x^{2} + 536x - 1272$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(109))$	8
Old forms	0
Cusp forms $S_{2}(\Gamma_0(109))$	8
Eisenstein series $E_{2}(\Gamma_0(109))$	1
Modular forms $M_{2}(\Gamma_0(109))$	9

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.16 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, 109]: [0, 9] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 8 = 1 x 8 (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(109, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 109:2 1000

← 108.2.a · 110.2.a →