Spectral Multiplicity for Maass Newforms of Non-Squarefree Level

We show that if a positive integer q has s(q) odd prime divisors p for which p(2) divides q, then a positive proportion of the Laplacian eigenvalues of Maass newforms of weight 0, level q, and principal character occur with multiplicity at least 2(s(q)). Consequently, the new part of the cuspidal spectrum of the Laplacian on Gamma(0)(q)\H cannot be simple for any odd non-squarefree integer q. This generalises work of Stromberg who proved this for q = 9 by different methods.