For odd square-free levels, the pullback of a Saito-Kurokawa lift decomposes into old and new pieces whose squared norms are exactly the central L-values L(f⊗sym^2 g, 1/2), with all other pieces vanishing.
Vanishing of certain spaces of elliptic modular forms and some applications
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Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass
For odd square-free levels, the pullback of a Saito-Kurokawa lift decomposes into old and new pieces whose squared norms are exactly the central L-values L(f⊗sym^2 g, 1/2), with all other pieces vanishing.