Atlas › Modular forms › Weight 2 › Level 31 › 31.2.a

Newspace 31.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(31))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
31.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} + 2q^{5} - 6q^{6} - 4q^{7} + O(q^{8})$	−	$+1$

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 $T$
31.2.a.a	$x^{2} + x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(31))$	2
Old forms	0
Cusp forms $S_{2}(\Gamma_0(31))$	2
Eisenstein series $E_{2}(\Gamma_0(31))$	1
Modular forms $M_{2}(\Gamma_0(31))$	3

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.1 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, 31]: [0, 3] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 2 = 1 x 2 (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(31, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 31:2 1000

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