  In the first half of the paper, we lay down a classical approach to the study
of Saito-Kurokawa (SK) lifts of (Hecke congruence) square-free level, including
the allied new-oldform theory. Our treatment of this relies on a novel idea of
computing ranks of certain matrices whose entries are $L^2$-norms of
eigenforms. For computing the $L^2$ norms we work with the Hecke algebra of
$\mathrm{GSp}(2)$.
  In the second half, we formulate precise conjectures on the $L^\infty$ size
of the space of SK lifts of square-free level, measured by the supremum of its
Bergman kernel, and prove bounds towards them using the results from the first
half. Here we rely on counting points on lattices, and on the geometric side of
the Bergman kernels of spaces of Jacobi forms underlying the SK lifts. Along
the way, we prove a non-trivial bound for the sup-norm of a Jacobi newform of
square-free level and also discuss about their size on average.
