Atlas › Modular forms › Weight 4 › Level 21 › 21.4.a

Newspace 21.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(21))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
21.4.a.a	1	$\mathbb{Q}$	$q - 3q^{2} - 3q^{3} + q^{4} - 18q^{5} + 9q^{6} + 7q^{7} + O(q^{8})$	+ −	$-1$
21.4.a.b	1	$\mathbb{Q}$	$q + 4q^{2} - 3q^{3} + 8q^{4} - 4q^{5} - 12q^{6} - 7q^{7} + O(q^{8})$	+ +	$+1$
21.4.a.c	2	$\mathbb{Q}(\sqrt{57})$	$2q - 3q^{2} + 6q^{3} + 17q^{4} + 6q^{5} - 9q^{6} + 14q^{7} + O(q^{8})$	− −	$+1$

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 $T$
21.4.a.a	$x + 21$
21.4.a.b	$x$
21.4.a.c	$x^{2} - 3x - 12$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(21))$	4
Old forms	2
Cusp forms $S_{4}(\Gamma_0(21))$	6
Eisenstein series $E_{4}(\Gamma_0(21))$	4
Modular forms $M_{4}(\Gamma_0(21))$	10

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(21)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 2, 3, 10] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 21:4 1000

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