Atlas › Modular forms › Weight 2 › Level 139 › 139.2.a

Newspace 139.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(139))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
139.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} - q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	−	$+1$	139a1 rank 0
139.2.a.b	3	degree 3	$3q - 2q^{2} - 2q^{3} - 8q^{5} - q^{6} + O(q^{8})$	+	$-1$
139.2.a.c	7	degree 7	$7q + q^{2} - 2q^{3} + 9q^{4} + 11q^{5} - 7q^{6} - 5q^{7} + O(q^{8})$	−	$+1$

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 $T$
139.2.a.a	$x - 3$
139.2.a.b	$x^{3} + 4x^{2} + 3x - 1$
139.2.a.c	$x^{7} + x^{6} - 21x^{5} + 2x^{4} + 97x^{3} + 4x^{2} - 140x - 72$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(139))$	11
Old forms	0
Cusp forms $S_{2}(\Gamma_0(139))$	11
Eisenstein series $E_{2}(\Gamma_0(139))$	1
Modular forms $M_{2}(\Gamma_0(139))$	12

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, 139]: [0, 12] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (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(139, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 139:2 1000

← 138.2.a · 140.2.a →