Atlas › Modular forms › Weight 2 › Level 251 › 251.2.a

Newspace 251.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(251))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
251.2.a.a	4	degree 4	$4q - 2q^{2} - 2q^{3} - 2q^{4} - 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+	$-1$
251.2.a.b	17	degree 17	$17q + 2q^{2} + 26q^{4} + 3q^{5} + q^{6} + 3q^{7} + O(q^{8})$	−	$+1$

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 $T$
251.2.a.a	$x^{4} + 4x^{3} - 3x - 1$
251.2.a.b	$x^{17} - 2x^{16} - 67x^{15} + 147x^{14} + 1702x^{13} - 3905x^{12} - 21240x^{11} + 49907x^{10} + 139820x^{9} - 338765x^{8} - 470516x^{7} + 1243843x^{6} + 649934x^{5} - 2319751x^{4} + 79259x^{3} + 1729437x^{2} - 623343x - 115664$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(251))$	21
Old forms	0
Cusp forms $S_{2}(\Gamma_0(251))$	21
Eisenstein series $E_{2}(\Gamma_0(251))$	1
Modular forms $M_{2}(\Gamma_0(251))$	22

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.17 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, 251]: [0, 22] (certified at ell = 2147482501)
T = [(2, 1), (3, 1)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 3 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(251, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 251:2 1000

← 250.2.a · 252.2.a →