  We analyze holomorphic Jacobi forms of weight one with level. One such form
plays an important role in umbral moonshine, leading to simplifications of the
statements of the umbral moonshine conjectures. We prove that non-zero
holomorphic Jacobi forms of weight one do not exist for many combinations of
index and level, and use this to establish a characterization of the
McKay--Thompson series of umbral moonshine in terms of Rademacher sums.
