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We show that the Dirichlet series associated to the Fourier coefficients of ahalf-integral weight Hecke eigenform at squarefree integers extendsanalytically to a holomorphic function in the half-plane $\res\textgreater{}\tfrac{1}{2}$. This exhibits a high fluctuation of thecoefficients at squarefree integers.

On Fourier coefficients of modular forms of half integral weight at squarefree integers