Multiplicity of an eigenvalue (reviewed)

The multiplicity of an eigenvalue $\lambda$ of the Laplace-Beltrami operator refers to the dimension of the eigenspace \( \mathcal{M}(\Gamma,\chi,\lambda) \) of Maass newforms on the group \( \Gamma \) of character \( \chi \) with the same Laplace eigenvalue \(\lambda\).

It is generally believed that these eigenspaces are one-dimensional unless there is a symmetry present. There are essentially three known cases where the dimension is greater than 1:

1. If the eigenvalue corresponds to an oldspace in the sense of Atkin and Lehner.

2. If the character \(\chi \) has quadratic characters as factors.

3. If \(\Gamma=\Gamma_{0}(N)\) with $9|N$

The above three cases are proven and the generalization of 3. to other square factors greater than 9 is conjectured.