We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane $\Re s>1/2$. This exhibits a high fluctuation of the coefficients at squarefree integers and improves a sign-change result in Lau et al. (Mathematika 62:866–883, 2016).