Atlas › Modular forms › Weight 8 › Level 21 › 21.8.a

Newspace 21.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(21))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
21.8.a.a	1	$\mathbb{Q}$	$q + 2q^{2} + 27q^{3} - 124q^{4} - 278q^{5} + 54q^{6} - 343q^{7} + O(q^{8})$	− +	$-1$
21.8.a.b	2	$\mathbb{Q}(\sqrt{1065})$	$2q - 9q^{2} - 54q^{3} + 317q^{4} - 360q^{5} + 243q^{6} + 686q^{7} + O(q^{8})$	+ −	$-1$
21.8.a.c	2	$\mathbb{Q}(\sqrt{67})$	$2q + 12q^{2} - 54q^{3} + 352q^{4} - 24q^{5} - 324q^{6} - 686q^{7} + O(q^{8})$	+ +	$+1$
21.8.a.d	3	degree 3	$3q - 3q^{2} + 81q^{3} + 225q^{4} - 114q^{5} - 81q^{6} + 1029q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
21.8.a.a	$x + 276$
21.8.a.b	$x^{2} + 369x - 62076$
21.8.a.c	$x^{2} + 12x - 21672$
21.8.a.d	$x^{3} + 117x^{2} - 189732x + 14892120$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(21))$	8
Old forms	8
Cusp forms $S_{8}(\Gamma_0(21))$	16
Eisenstein series $E_{8}(\Gamma_0(21))$	4
Modular forms $M_{8}(\Gamma_0(21))$	20

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(21)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.262 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, 3, 7, 21]: [1, 3, 5, 20] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(21, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 21:8 1000

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