Atlas › Modular forms › Weight 2 › Level 257 › 257.2.a

Newspace 257.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(257))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
257.2.a.a	7	degree 7	$7q - 3q^{2} - 5q^{3} + q^{4} - q^{5} - 2q^{6} - 18q^{7} + O(q^{8})$	+	$-1$
257.2.a.b	14	degree 14	$14q + 2q^{2} + 3q^{3} + 18q^{4} - q^{5} - 4q^{6} + 24q^{7} + O(q^{8})$	−	$+1$

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 $T$
257.2.a.a	$x^{7} + 8x^{6} + 15x^{5} - 8x^{4} - 42x^{3} - 30x^{2} - 4x + 1$
257.2.a.b	$x^{14} - 5x^{13} - 34x^{12} + 192x^{11} + 352x^{10} - 2662x^{9} - 398x^{8} + 15681x^{7} - 11622x^{6} - 32938x^{5} + 45747x^{4} + 1133x^{3} - 21671x^{2} + 4763x + 1373$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(257))$	21
Old forms	0
Cusp forms $S_{2}(\Gamma_0(257))$	21
Eisenstein series $E_{2}(\Gamma_0(257))$	1
Modular forms $M_{2}(\Gamma_0(257))$	22

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.168 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, 257]: [0, 22] (certified at ell = 2147483137)
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(257, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 257:2 1000

← 256.2.a · 258.2.a →