Non-linear additive twists of GL(2) ×GL(2) cusp-form coefficients over primes

Let [Formula: see text] and [Formula: see text] be holomorphic or Maass cusp forms for [Formula: see text] with normalized Fourier coefficients [Formula: see text] and [Formula: see text], respectively. In this paper, we study the non-linear additive twisted sum of [Formula: see text] forms [Formula: see text] where [Formula: see text]. Previously, [Formula: see text] and [Formula: see text] cases have been investigated extensively.