Atlas › Modular forms › Weight 10 › Level 21 › 21.10.a

Newspace 21.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(21))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
21.10.a.a	1	$\mathbb{Q}$	$q - 24q^{2} + 81q^{3} + 64q^{4} - 144q^{5} - 1944q^{6} + 2401q^{7} + O(q^{8})$	− −	$-1$
21.10.a.b	2	$\mathbb{Q}(\sqrt{2353})$	$2q + 9q^{2} - 162q^{3} + 193q^{4} + 1170q^{5} - 729q^{6} - 4802q^{7} + O(q^{8})$	+ +	$-1$
21.10.a.c	2	$\mathbb{Q}(\sqrt{345})$	$2q + 30q^{2} - 162q^{3} + 116q^{4} + 1128q^{5} - 2430q^{6} + 4802q^{7} + O(q^{8})$	+ −	$+1$
21.10.a.d	3	degree 3	$3q - 13q^{2} + 243q^{3} + 1677q^{4} + 2398q^{5} - 1053q^{6} - 7203q^{7} + O(q^{8})$	− +	$+1$

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 $T$
21.10.a.a	$x + 168$
21.10.a.b	$x^{2} - 1179x - 2453148$
21.10.a.c	$x^{2} - 1158x - 1403904$
21.10.a.d	$x^{3} - 2385x^{2} + 1537722x - 136361232$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(21))$	8
Old forms	14
Cusp forms $S_{10}(\Gamma_0(21))$	22
Eisenstein series $E_{10}(\Gamma_0(21))$	4
Modular forms $M_{10}(\Gamma_0(21))$	26

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(21)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 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, 4, 7, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(21, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 21:10 1000

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