Atlas › Modular forms › Weight 2 › Level 110 › 110.2.a

Newspace 110.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(110))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
110.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 5q^{7} + O(q^{8})$	+ + −	$+1$	110c1 rank 0
110.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$	110a1 rank 0
110.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	110b1 rank 0
110.2.a.d	2	$\mathbb{Q}(\sqrt{33})$	$2q - 2q^{2} - q^{3} + 2q^{4} + 2q^{5} + q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
110.2.a.a	$x - 16$
110.2.a.b	$x - 8$
110.2.a.c	$x + 2$
110.2.a.d	$x^{2} - 2x - 32$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(110))$	5
Old forms	10
Cusp forms $S_{2}(\Gamma_0(110))$	15
Eisenstein series $E_{2}(\Gamma_0(110))$	7
Modular forms $M_{2}(\Gamma_0(110))$	22

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.093 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]: [0, 1, 1, 3, 2, 5, 8, 22] (certified at ell = 2147482921)
T = [(3, 1), (7, 3)]; h of degree 5 = 1 x 5 (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(110, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 110:2 1000

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