Atlas › Modular forms › Weight 6 › Level 110 › 110.6.a

Newspace 110.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(110))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
110.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} + 2q^{3} + 16q^{4} + 25q^{5} - 8q^{6} - 40q^{7} + O(q^{8})$	+ − −	$-1$
110.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} + 12q^{3} + 16q^{4} - 25q^{5} - 48q^{6} + 54q^{7} + O(q^{8})$	+ + +	$-1$
110.6.a.c	1	$\mathbb{Q}$	$q + 4q^{2} - 12q^{3} + 16q^{4} - 25q^{5} - 48q^{6} + 110q^{7} + O(q^{8})$	− + −	$-1$
110.6.a.d	1	$\mathbb{Q}$	$q + 4q^{2} - 2q^{3} + 16q^{4} + 25q^{5} - 8q^{6} - 204q^{7} + O(q^{8})$	− − +	$-1$
110.6.a.e	2	$\mathbb{Q}(\sqrt{889})$	$2q - 8q^{2} + 21q^{3} + 32q^{4} - 50q^{5} - 84q^{6} + 35q^{7} + O(q^{8})$	+ + −	$+1$
110.6.a.f	2	$\mathbb{Q}(\sqrt{1321})$	$2q + 8q^{2} - 3q^{3} + 32q^{4} - 50q^{5} - 12q^{6} + 237q^{7} + O(q^{8})$	− + +	$+1$
110.6.a.g	3	degree 3	$3q - 12q^{2} - 7q^{3} + 48q^{4} + 75q^{5} + 28q^{6} + 273q^{7} + O(q^{8})$	+ − +	$+1$
110.6.a.h	3	degree 3	$3q + 12q^{2} + 25q^{3} + 48q^{4} + 75q^{5} + 100q^{6} + 159q^{7} + O(q^{8})$	− − −	$+1$

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 $T$
110.6.a.a	$x + 38$
110.6.a.b	$x - 66$
110.6.a.c	$x - 98$
110.6.a.d	$x + 206$
110.6.a.e	$x^{2} - 56x - 2772$
110.6.a.f	$x^{2} - 234x + 1800$
110.6.a.g	$x^{3} - 266x^{2} - 24000x + 7336800$
110.6.a.h	$x^{3} - 184x^{2} - 5348x + 30048$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(110))$	14
Old forms	72
Cusp forms $S_{6}(\Gamma_0(110))$	86
Eisenstein series $E_{6}(\Gamma_0(110))$	8
Modular forms $M_{6}(\Gamma_0(110))$	94

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(110)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 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, 11, 22, 55, 110]: [1, 2, 3, 9, 6, 17, 32, 94] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 14 = 1 x 14 (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(110, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 110:6 1000

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