  Recently, the Nash-style convex integration has been becoming the main scheme
for the mathematical study of turbulence, and the main building block of it has
been either Beltrami flow (finite mode) or Mikado flow (compactly supported in
the physical side). On the other hand, in physics, it is observed that
turbulence is composed of a hierarchy of scale-by-scale vortex stretching. Thus
our mathematical motivation in this study is to find another type of building
blocks accompanied by vortex stretching and scale locality (possibly finitely
many Fourier modes). In this paper, we give a complete list of solutions to the
3D Euler equations with finitely many Fourier modes, which is an extension of
the corresponding 2D result by Elgindi-Hu-\v{S}ver\'ak (2017). In particular,
we show that there is no 3D Euler flows with finitely many Fourier modes,
except for stationary 2D-like flows and Beltrami flows. We also discuss the
case when viscosity and Coriolis effect are present.
