Atlas › Modular forms › Weight 2 › Level 317 › 317.2.a

Newspace 317.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(317))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
317.2.a.a	11	degree 11	$11q - 3q^{2} - 11q^{3} + 7q^{4} - 4q^{5} - 6q^{6} - 20q^{7} + O(q^{8})$	+	$-1$
317.2.a.b	15	degree 15	$15q + q^{2} + 11q^{3} + 15q^{4} + 2q^{5} + 20q^{7} + O(q^{8})$	−	$+1$

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 $T$
317.2.a.a	$x^{11} + 14x^{10} + 66x^{9} + 93x^{8} - 163x^{7} - 613x^{6} - 533x^{5} + 26x^{4} + 179x^{3} + 14x^{2} - 14x + 1$
317.2.a.b	$x^{15} - 12x^{14} + 19x^{13} + 283x^{12} - 1124x^{11} - 1420x^{10} + 12920x^{9} - 8624x^{8} - 50379x^{7} + 77325x^{6} + 45318x^{5} - 129347x^{4} + 13755x^{3} + 58453x^{2} - 8624x - 7517$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(317))$	26
Old forms	0
Cusp forms $S_{2}(\Gamma_0(317))$	26
Eisenstein series $E_{2}(\Gamma_0(317))$	1
Modular forms $M_{2}(\Gamma_0(317))$	27

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.175 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, 317]: [0, 27] (certified at ell = 2147479753)
T = [(2, 1), (3, 1)]; h of degree 26 = 1 x 26 (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(317, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 317:2 1000

← 316.2.a · 318.2.a →