On explicit Fourier expansions of theta lifts to ${\rm SO}(3,n+1)$ arising from elliptic newforms of level one

Abstract

Using degenerate Whittaker functions and explicit computations of Eisenstein series, we obtain explicit formulas for the Fourier expansions of theta lifts to the special orthogonal group $G={\rm SO}(3,n+1)$ over $\mathbb{Q}$, where $n\ge 3$ and $G$ splits at all finite places. The theta lifts in question are Hecke eigen, non-cuspidal, square-integrable automorphic forms of weight $l$ ($l\ge n+2$, even), arising from elliptic newforms for $\SL_2(\Z)$ of weight $l-\frac{n-2}{2}$ when $n$ is even and $2l-n+1$ when $n$ is odd.