Fourier expansion of the Saito-Kurokawa lift

Question

As is well known, the Saito-Kurokawa lifts maps (classical) cusp forms $f$ to Siegel (genus 2) cusp forms $SK(f)$.

Is there an explicit formula for the Fourier expansion of a Saito-Kurokawa lift?

By explicit I mean something expressing the coefficients of $SK(f)$ in terms of the ones of $f$. Also, I am interested in the level $1$ case (the whole modular groups).

I know how the lift is constructed (passing via Jacobi forms and half-integral modular forms, as is well explained in Van der Geer, The 1-2-3 of modular forms - https://link.springer.com/book/10.1007%2F978-3-540-74119-0) but I could not manage to derive an explicit formula nor could I find anything online.

I know about the relation between eigenvalues of $f$ and $SK(f)$ - when they are eigenforms - but as far as I remember knowing the eigenvalues of Siegel modular forms is not sufficient to compute the Fourier coefficients (while the converse is true).

I would be grateful to be directed to any reference that could potentially help me.

Answer 1

Let's say $f$ is of weight $2k-2$ and $g$ is the associated form of weight $k - 1/2$. One can relate the Fourier coefficients of $g$ to those of the associated Jacobi form $J$, the Fourier coefficients of $J$ to those of the Saito-Kurokawa lift $F$ of $f$. See

Agarwal, Mahesh; Brown, Jim. Saito-Kurokawa lifts of square-free level. Kyoto J. Math. 55 (2015), no. 3, 641–662.

and references therein for details. Hence one can relate Fourier coefficients of $g$ to those of $F$, though I do not know the details enough to know if you can get a formula for individual Fourier coefficients of $F$ in terms of those of $g$.

From another point of view, one has an explicit relation between square sums of Fourier coefficients of $F$ and squares of Fourier coefficients of $g$. Namely, Bocherer conjectured that twisted central spinor $L$-values of $F$ are essentially squares of Bessel periods, and he proved this for Saito-Kurokawa lifts. Here the $L$-values $L(1/2,F,\chi)$ can be rewritten in terms of twisted $L$-values $L(1/2,f,\chi)$ of $f$, which are essentially squares of Fourier coefficients of $g$ by a formula of Waldspurger. (Here $\chi$ is a quadratic character.) The Bessel periods are sums of Fourier coefficients indexed by the same discriminant. Bocherer's work was unpublished, but you can see Dickson, Pitale, Saha and Schmidt for more details.

Moreover, this connection between Fourier coefficients of $g$ and $L$-values $L(1/2,f,\chi)$ means there is no simple connection between Fourier coefficients of $f$ and those of $g$ or $F$.

Finally, regarding your penultimate paragraph: it is true that the Hecke eigenvalues (though not just for $T_p$'s) of a holomorphic Siegel modular newform determine the form. Thus the eigenvalues determine (abstractly) the Fourier coefficients of such forms, but indeed this is different from being able to compute the Fourier coefficients given the eigenvalues.