Atlas › Modular forms › Weight 12 › Level 21 › 21.12.a

Newspace 21.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(21))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
21.12.a.a	1	$\mathbb{Q}$	$q - 62q^{2} + 243q^{3} + 1796q^{4} - 3310q^{5} - 15066q^{6} - 16807q^{7} + O(q^{8})$	− +	$-1$
21.12.a.b	1	$\mathbb{Q}$	$q + 8q^{2} + 243q^{3} - 1984q^{4} + 4390q^{5} + 1944q^{6} - 16807q^{7} + O(q^{8})$	− +	$-1$
21.12.a.c	3	degree 3	$3q - 68q^{2} - 729q^{3} + 1920q^{4} + 3326q^{5} + 16524q^{6} - 50421q^{7} + O(q^{8})$	+ +	$+1$
21.12.a.d	3	degree 3	$3q - 33q^{2} - 729q^{3} + 765q^{4} + 3102q^{5} + 8019q^{6} + 50421q^{7} + O(q^{8})$	+ −	$-1$
21.12.a.e	4	degree 4	$4q + 45q^{2} + 972q^{3} + 4801q^{4} + 13356q^{5} + 10935q^{6} + 67228q^{7} + O(q^{8})$	− −	$+1$

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 $T$
21.12.a.a	$x + 3372$
21.12.a.b	$x - 4398$
21.12.a.c	$x^{3} - 3258x^{2} - 154567152x + 817858898640$
21.12.a.d	$x^{3} - 3069x^{2} - 37748730x + 60897667320$
21.12.a.e	$x^{4} - 13401x^{3} - 70137066x^{2} + 946011785040x + 294712626140496$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(21))$	12
Old forms	16
Cusp forms $S_{12}(\Gamma_0(21))$	28
Eisenstein series $E_{12}(\Gamma_0(21))$	4
Modular forms $M_{12}(\Gamma_0(21))$	32

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(21)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.534 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]: [2, 5, 9, 32] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 6 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, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 21:12 1000

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