Atlas › Modular forms › Weight 2 › Level 50 › 50.2.a

Newspace 50.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(50))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
50.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + 2q^{7} + O(q^{8})$	+			50a1 rank 0
50.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - 2q^{7} + O(q^{8})$	−			50b1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
50.2.a.a	$x - 3$
50.2.a.b	$x + 3$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(50))$	2
Old forms	0
Cusp forms $S_{2}(\Gamma_0(50))$	2
Eisenstein series $E_{2}(\Gamma_0(50))$	11
Modular forms $M_{2}(\Gamma_0(50))$	13

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.089 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, 2, 5, 10, 25, 50]: [0, 1, 1, 3, 3, 9] (certified at ell = 2147482921)
T = [(3, 1), (7, 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(50, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 50:2 1000

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