Atlas › Modular forms › Weight 8 › Level 19 › 19.8.a

Newspace 19.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(19))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
19.8.a.a	4	degree 4	$4q - 9q^{2} - 14q^{3} + 37q^{4} - 222q^{5} - 603q^{6} - 1246q^{7} + O(q^{8})$	−	$-1$
19.8.a.b	6	degree 6	$6q + 15q^{2} + 40q^{3} + 357q^{4} + 219q^{5} + 925q^{6} + 2105q^{7} + O(q^{8})$	+	$+1$

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 $T$
19.8.a.a	$x^{4} + 23x^{3} - 1332x^{2} - 12312x + 415872$
19.8.a.b	$x^{6} - 55x^{5} - 9494x^{4} + 463630x^{3} + 23057000x^{2} - 985214600x - 3813684864$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(19))$	10
Old forms	0
Cusp forms $S_{8}(\Gamma_0(19))$	10
Eisenstein series $E_{8}(\Gamma_0(19))$	2
Modular forms $M_{8}(\Gamma_0(19))$	12

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.265 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, 19]: [1, 12] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (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(19, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 19:8 1000

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