Atlas › Modular forms › Weight 2 › Level 131 › 131.2.a

Newspace 131.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(131))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
131.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - 2q^{5} - q^{7} + O(q^{8})$	+	$-1$		131a1 rank 1
131.2.a.b	10	degree 10	$10q + q^{3} + 16q^{4} + 4q^{5} - 4q^{6} + q^{7} + O(q^{8})$	−	$+1$

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 $T$
131.2.a.a	$x + 1$
131.2.a.b	$x^{10} - x^{9} - 36x^{8} + 70x^{7} + 313x^{6} - 924x^{5} + 401x^{4} + 387x^{3} - 270x^{2} + 36x + 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(131))$	11
Old forms	0
Cusp forms $S_{2}(\Gamma_0(131))$	11
Eisenstein series $E_{2}(\Gamma_0(131))$	1
Modular forms $M_{2}(\Gamma_0(131))$	12

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.161 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, 131]: [0, 12] (certified at ell = 2147482481)
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(131, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 131:2 1000

← 130.2.a · 132.2.a →