Atlas › Modular forms › Weight 6 › Level 19 › 19.6.a

Newspace 19.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(19))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
19.6.a.a	1	$\mathbb{Q}$	$q - 6q^{2} + 4q^{3} + 4q^{4} + 54q^{5} - 24q^{6} + 248q^{7} + O(q^{8})$	−	$+1$
19.6.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} - 28q^{4} - 24q^{5} + 2q^{6} - 167q^{7} + O(q^{8})$	+	$-1$
19.6.a.c	2	$\mathbb{Q}(\sqrt{177})$	$2q - 7q^{2} - 7q^{3} + 49q^{4} - 133q^{5} - 241q^{6} + 72q^{7} + O(q^{8})$	+	$-1$
19.6.a.d	4	degree 4	$4q + 9q^{2} + 6q^{3} + 97q^{4} + 90q^{5} + 369q^{6} - 190q^{7} + O(q^{8})$	−	$+1$

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 $T$
19.6.a.a	$x + 2$
19.6.a.b	$x + 3$
19.6.a.c	$x^{2} + 14x - 128$
19.6.a.d	$x^{4} - 15x^{3} - 1278x^{2} + 25974x - 82176$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(19))$	8
Old forms	0
Cusp forms $S_{6}(\Gamma_0(19))$	8
Eisenstein series $E_{6}(\Gamma_0(19))$	2
Modular forms $M_{6}(\Gamma_0(19))$	10

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.211 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, 10] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 19:6 1000

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