Low moments of random multiplicative functions twisted by Fourier coefficients of modular forms

Let $\lambda (n)$ denote the Fourier coefficients of a fixed modular form and $h(n)$ a Steinhaus or Rademacher random multiplicative function. In this paper, we determine the order of magnitude of $$\text{E}\mathrm{\mid }\sum _{n\le x}h(n)\lambda (n){\mathrm{\mid }}^{2q}$$ for real $x$, $q$ with $0\le q\le 1$.