Atlas › Modular forms › Weight 2 › Level 39 › 39.2.a

Newspace 39.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(39))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
39.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$		39a1 rank 0
39.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
39.2.a.a	$x - 3$
39.2.a.b	$x^{2} + 2x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(39))$	3
Old forms	0
Cusp forms $S_{2}(\Gamma_0(39))$	3
Eisenstein series $E_{2}(\Gamma_0(39))$	3
Modular forms $M_{2}(\Gamma_0(39))$	6

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.088 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, 3, 13, 39]: [0, 1, 1, 6] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 3 = 1 x 3 (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(39, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 39:2 1000

← 38.2.a · 40.2.a →