By the multiplicity of a Laplace eigenvalue we mean the dimension of the space \( \mathcal{M}(\Gamma,\chi,\lambda) \) of Maass waveforms on the group \( \Gamma \) and character \( \chi \) and 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.
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 Last edited by Nathan Ryan on 20190501 13:29:08
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 20190501 13:29:08 by Nathan Ryan (Reviewed)
 20190429 23:47:44 by Nathan Ryan
 20120401 23:19:46 by Fredrik Strömberg