Atlas › Modular forms › Weight 8 › Level 31 › 31.8.a

Newspace 31.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(31))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
31.8.a.a	7	degree 7	$7q - 17q^{2} - 14q^{3} + 229q^{4} - 430q^{5} - 528q^{6} - 832q^{7} + O(q^{8})$	−	$-1$
31.8.a.b	10	degree 10	$10q + 15q^{2} + 40q^{3} + 613q^{4} + 570q^{5} + 768q^{6} + 540q^{7} + O(q^{8})$	+	$+1$

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 $T$
31.8.a.a	$x^{7} + 31x^{6} - 8266x^{5} - 132930x^{4} + 15588628x^{3} + 111486640x^{2} - 8449179712x - 4467019008$
31.8.a.b	$x^{10} - 55x^{9} - 15896x^{8} + 695035x^{7} + 86166363x^{6} - 2886863810x^{5} - 202197440780x^{4} + 4611097028760x^{3} + 220580468861120x^{2} - 2496232761718400x - 93064067996201984$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(31))$	17
Old forms	0
Cusp forms $S_{8}(\Gamma_0(31))$	17
Eisenstein series $E_{8}(\Gamma_0(31))$	2
Modular forms $M_{8}(\Gamma_0(31))$	19

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.266 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, 31]: [1, 19] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 4 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(31, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 31:8 1000

← 30.8.a · 32.8.a →