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We investigate the growth of Fourier coefficients of Siegel paramodular formsbuilt by exponentially lifting weak Jacobi forms, focusing on terms with largenegative discriminant. To this end we implement a method based on deformingcontours that expresses the coefficients of all such terms as residues. We findthat there are two types of weak Jacobi forms, leading to two different growthbehaviors: the more common type leads to fast, exponential growth, whereas asecond type leads to slower growth, akin to the growth seen in ratios of thetafunctions. We give a simple criterion to distinguish between the two types, andgive a simple closed form expression for the coefficients in the slow growingcase. In a companion article [1], we provide physical applications of theseresults to symmetric product orbifolds and holography.

Siegel Paramodular Forms from Exponential Lifts: Slow versus Fast Growth