Atlas › Modular forms › Weight 10 › Level 10 › 10.10.a

Newspace 10.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(10))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
10.10.a.a	1	$\mathbb{Q}$	$q - 16q^{2} - 204q^{3} + 256q^{4} + 625q^{5} + 3264q^{6} + 5432q^{7} + O(q^{8})$	+ −	$+1$
10.10.a.b	1	$\mathbb{Q}$	$q - 16q^{2} + 46q^{3} + 256q^{4} - 625q^{5} - 736q^{6} - 10318q^{7} + O(q^{8})$	+ +	$-1$
10.10.a.c	1	$\mathbb{Q}$	$q + 16q^{2} + 174q^{3} + 256q^{4} - 625q^{5} + 2784q^{6} + 4658q^{7} + O(q^{8})$	− +	$+1$

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 $T$
10.10.a.a	$x - 5228$
10.10.a.b	$x + 10272$
10.10.a.c	$x - 4832$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(10))$	3
Old forms	8
Cusp forms $S_{10}(\Gamma_0(10))$	11
Eisenstein series $E_{10}(\Gamma_0(10))$	4
Modular forms $M_{10}(\Gamma_0(10))$	15

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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]: [1, 3, 5, 15] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 2 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, 10)

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

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

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