Atlas › Modular forms › Weight 2 › Level 220 › 220.2.a

Newspace 220.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(220))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
220.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} - 4q^{7} + O(q^{8})$	− +			220a1 rank 1
220.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + O(q^{8})$	− −			220b1 rank 0

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 $T$
220.2.a.a	$x + 14$
220.2.a.b	$x - 2$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(220))$	2
Old forms	29
Cusp forms $S_{2}(\Gamma_0(220))$	31
Eisenstein series $E_{2}(\Gamma_0(220))$	11
Modular forms $M_{2}(\Gamma_0(220))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(220)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(110))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.228 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, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220]: [0, 1, 2, 1, 3, 2, 6, 5, 9, 8, 22, 42] (certified at ell = 2147482921)
T = [(3, 1), (7, 3)]; h of degree 2 = 1 x 2 (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(220, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 220:2 1000

← 219.2.a · 221.2.a →