Atlas › Modular forms › Weight 4 › Level 110 › 110.4.a

Newspace 110.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(110))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
110.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 7q^{3} + 4q^{4} + 5q^{5} + 14q^{6} - 35q^{7} + O(q^{8})$	+ − −	$+1$
110.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} - 30q^{7} + O(q^{8})$	+ + −	$-1$
110.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{3} + 4q^{4} + 5q^{5} - 8q^{6} + 20q^{7} + O(q^{8})$	+ − −	$+1$
110.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - 8q^{3} + 4q^{4} - 5q^{5} - 16q^{6} + 26q^{7} + O(q^{8})$	− + −	$+1$
110.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} - 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} - 22q^{7} + O(q^{8})$	− + +	$-1$
110.4.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 4q^{4} + 5q^{5} + 2q^{6} + 23q^{7} + O(q^{8})$	− − +	$+1$
110.4.a.g	1	$\mathbb{Q}$	$q + 2q^{2} + 7q^{3} + 4q^{4} - 5q^{5} + 14q^{6} + 11q^{7} + O(q^{8})$	− + −	$+1$
110.4.a.h	1	$\mathbb{Q}$	$q + 2q^{2} + 8q^{3} + 4q^{4} + 5q^{5} + 16q^{6} - 12q^{7} + O(q^{8})$	− − +	$+1$
110.4.a.i	2	$\mathbb{Q}(\sqrt{177})$	$2q - 4q^{2} + 7q^{3} + 8q^{4} - 10q^{5} - 14q^{6} + 27q^{7} + O(q^{8})$	+ + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{13} + 7T_{17}$ 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 $T$
110.4.a.a	$x - 725$
110.4.a.b	$x + 790$
110.4.a.c	$x + 100$
110.4.a.d	$x + 58$
110.4.a.e	$x + 310$
110.4.a.f	$x - 845$
110.4.a.g	$x + 13$
110.4.a.h	$x + 800$
110.4.a.i	$x^{2} - 641x - 423014$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(110))$	10
Old forms	40
Cusp forms $S_{4}(\Gamma_0(110))$	50
Eisenstein series $E_{4}(\Gamma_0(110))$	8
Modular forms $M_{4}(\Gamma_0(110))$	58

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.154 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, 7, 4, 11, 20, 58] (certified at ell = 2147482921)
T = [(3, 1), (7, 3), (13, 5), (17, 7)]; h of degree 10 = 1 x 10 (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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 110:4 1000

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