  In the present work, we establish the approximation of nonlinear stochastic
partial differential equation (SPDE) driven by cylindrical {\alpha}-stable
L\'evy processes via modulation or amplitude equations.
  We study SPDEs with a cubic nonlinearity, where the deterministic equation is
close to a change of stability of the trivial solution. The natural separation
of time-scales close to this bifurcation allows us to obtain an amplitude
equation describing the essential dynamics of the bifurcating pattern, thus
reducing the original infinite dimensional dynamics to a simpler
finite-dimensional effective dynamics. In the presence of a multiplicative
stable L\'evy noise that preserves the constant trivial solution we study the
impact of noise on the approximation.
  In contrast to Gaussian noise, where non-dominant pattern are uniformly small
in time due to averaging effects, large jumps in the L\'evy noise might lead to
large error terms, and thus new estimates are needed to take this into account.
