Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the L²-mass

We obtain the full spectral decomposition of the pullback of a Saito-Kurokawa (SK) newform F of odd, square-free level; and show that the projections onto the elements g ⊗ g of an arithmetically orthogonalized old-basis are either zero or whose squares are given by the certain GL(3)× GL(2) central L-values L(f⊗ sym² g, 1/2), where F is the lift of the GL(2) newform f and g is the newform underlying g. Based on this, we work out a conjectural formula for the L²-mass of the pullback of F via the CFKRS heuristics, which becomes a weighted average (over g) of the central L-values. We show that on average over f, the main term predicted by the above heuristics matches with the actual main term. We also provide several results and sufficient conditions that ensure the non-vanishing of the pullbacks.