Atlas › Modular forms › Weight 2 › Level 180 › 180.2.a

Newspace 180.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(180))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
180.2.a.a	1	$\mathbb{Q}$	$q + q^{5} + 2q^{7} + O(q^{8})$	−			180a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
180.2.a.a	$x - 18$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(180))$	1
Old forms	24
Cusp forms $S_{2}(\Gamma_0(180))$	25
Eisenstein series $E_{2}(\Gamma_0(180))$	23
Modular forms $M_{2}(\Gamma_0(180))$	48

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(180)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(90))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.148 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, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180]: [0, 1, 1, 2, 1, 3, 2, 3, 5, 4, 5, 6, 10, 9, 8, 18, 22, 42] (certified at ell = 2147483629)
T = [(7, 1), (11, 3), (13, 5), (17, 7), (19, 9)]; h of degree 1 = 1 x 1 (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(180, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 180:2 1000

← 179.2.a · 181.2.a →