Atlas › Modular forms › Weight 12 › Level 10 › 10.12.a

Newspace 10.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(10))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
10.12.a.a	1	$\mathbb{Q}$	$q - 32q^{2} - 12q^{3} + 1024q^{4} + 3125q^{5} + 384q^{6} - 14176q^{7} + O(q^{8})$	+ −	$-1$
10.12.a.b	1	$\mathbb{Q}$	$q - 32q^{2} + 738q^{3} + 1024q^{4} - 3125q^{5} - 23616q^{6} + 25574q^{7} + O(q^{8})$	+ +	$+1$
10.12.a.c	1	$\mathbb{Q}$	$q + 32q^{2} - 318q^{3} + 1024q^{4} - 3125q^{5} - 10176q^{6} - 70714q^{7} + O(q^{8})$	− +	$-1$
10.12.a.d	2	$\mathbb{Q}(\sqrt{1969})$	$2q + 64q^{2} + 604q^{3} + 2048q^{4} + 6250q^{5} + 19328q^{6} + 14092q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
10.12.a.a	$x + 14188$
10.12.a.b	$x - 26312$
10.12.a.c	$x + 71032$
10.12.a.d	$x^{2} - 14696x - 6045181296$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(10))$	5
Old forms	10
Cusp forms $S_{12}(\Gamma_0(10))$	15
Eisenstein series $E_{12}(\Gamma_0(10))$	4
Modular forms $M_{12}(\Gamma_0(10))$	19

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(10)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.332 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, 2, 5, 10]: [2, 4, 7, 19] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 5 = 1 x 5 (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(10, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 10:12 1000

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