A classical problem in harmonic analysis is to understand how the mass of Laplace eigenfunctions distributes as their parameters tend to infinity. In this talk, I will consider this problem on arithmetic hyperbolic surfaces obtained as quotients of the upper half plane by congruence subgroups of SL_2(Z). I will focus on the depth aspect, where the level of the surface grows through powers of a fixed prime. More precisely, let f be an L^2-normalized newform for GL(2) of level p^n, with p a fixed odd prime, and n tending to infinity. Assuming that f has trivial central character and bounded spectral parameters, I will discuss the optimal bound for its L^4-norm and the analogies with the spectral aspect.