Atlas › Modular forms › Weight 2 › Level 51 › 51.2.a

Newspace 51.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(51))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
51.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	− +	$+1$		51a1 rank 0
51.2.a.b	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} - 2q^{3} + 5q^{4} + 3q^{5} + q^{6} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
51.2.a.a	$x - 9$
51.2.a.b	$x^{2} - 8x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(51))$	3
Old forms	2
Cusp forms $S_{2}(\Gamma_0(51))$	5
Eisenstein series $E_{2}(\Gamma_0(51))$	3
Modular forms $M_{2}(\Gamma_0(51))$	8

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(51)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.087 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, 17, 51]: [0, 1, 2, 8] (certified at ell = 2147483489)
T = [(2, 1), (5, 3)]; 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(51, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 51:2 1000

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