  We study the probabilistic behavior of sums of Fourier coefficients in
arithmetic progressions. We prove a result analogous to previous work of
Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of
half-integral weight holomorphic cusp forms and for prime power modulus. We
actually show that these sums follow in a suitable range a mixed Gaussian
distribution which comes from the asymptotic mixed distribution of Sali\'e
sums.
