The multiplicity of an eigenvalue $\lambda$ of the LaplaceBeltrami 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 onedimensional unless there is a symmetry present. There are essentially three known cases where the dimension is greater than 1:

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

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

If \(\Gamma=\Gamma_{0}(N)\) with $9N$
The above three cases are proven and the generalization of 3. to other square factors greater than 9 is conjectured.
 Review status: reviewed
 Last edited by Andrew Sutherland on 20200723 16:03:15
Not referenced anywhere at the moment.
 20200723 16:03:15 by Andrew Sutherland (Reviewed)
 20190501 13:29:08 by Nathan Ryan (Reviewed)
 20190429 23:47:44 by Nathan Ryan
 20120401 23:19:46 by Fredrik Strömberg