Atlas › Modular forms › Weight 2 › Level 37 › 37.2.a

Newspace 37.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(37))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
37.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} + 6q^{6} - q^{7} + O(q^{8})$	+	$-1$		37a1 rank 1
37.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{7} + O(q^{8})$	−	$+1$		37b1 rank 0

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 $T$
37.2.a.a	$x + 5$
37.2.a.b	$x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(37))$	2
Old forms	0
Cusp forms $S_{2}(\Gamma_0(37))$	2
Eisenstein series $E_{2}(\Gamma_0(37))$	1
Modular forms $M_{2}(\Gamma_0(37))$	3

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.173 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, 37]: [0, 3] (certified at ell = 2147483629)
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(37, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 37:2 1000

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