Atlas › Modular forms › Weight 6 › Level 10 › 10.6.a

Newspace 10.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(10))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
10.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 26q^{3} + 16q^{4} - 25q^{5} + 104q^{6} - 22q^{7} + O(q^{8})$	+ +	$-1$
10.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} + 24q^{3} + 16q^{4} + 25q^{5} - 96q^{6} - 172q^{7} + O(q^{8})$	+ −	$+1$
10.6.a.c	1	$\mathbb{Q}$	$q + 4q^{2} + 6q^{3} + 16q^{4} - 25q^{5} + 24q^{6} - 118q^{7} + O(q^{8})$	− +	$+1$

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 $T$
10.6.a.a	$x + 48$
10.6.a.b	$x + 148$
10.6.a.c	$x + 112$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(10))$	3
Old forms	2
Cusp forms $S_{6}(\Gamma_0(10))$	5
Eisenstein series $E_{6}(\Gamma_0(10))$	4
Modular forms $M_{6}(\Gamma_0(10))$	9

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(10)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.105 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, 2, 3, 9] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 3 = 1 x 3 (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(10, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 10:6 1000

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