A Maass Lifting of ${\mathrm{\Theta }}^3$ and Class Numbers of Real and imaginary Quadratic Fields

We give an explicit construct of a harmonic weak Maass form $F_{\mathrm{\Theta }}$ that is a “lift” of ${\mathrm{\Theta }}^3$, where $\mathrm{\Theta }$ is the classical Jacobi theta function. Just as the Fourier coefficients of ${\mathrm{\Theta }}^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the “holomorphic part” of $F_{\mathrm{\Theta }}$ are associated to class numbers of real quadratic fields.