Summary: In type II string theory compactified on a \(d\)-dimensional torus \(T^{d}\) down to \(D = 10-d\) dimensions, the \(\mathcal{R}^{4}\) and \(\nabla^{4}\mathcal{R}^{4}\) four-graviton couplings are known exactly, for all values of the moduli, in terms of certain Eisenstein series of the U-duality group \(E_{d}(\mathbb Z)\). In the limit where one circle in the torus becomes large, these couplings are expected to reduce to their counterpart in dimension \(D +1\), plus threshold effects and exponentially suppressed corrections corresponding to BPS black holes in dimension \(D + 1\) whose worldline winds around the circle. By combining the weak coupling and large radius limits, we determine these exponentially suppressed corrections exactly, and demonstrate that the contributions of \(1/4\)-BPS black holes to the \(\nabla^{4}\mathcal{R}^{4}\) coupling are proportional to the appropriate helicity supertrace. Mathematically, our results provide the complete Fourier expansion of the next-to-minimal theta series of \(E_{d\;+\;1}(\mathbb Z)\) with respect to the maximal parabolic subgroup with Levi component \(E_{d}\) for \(d \leq 6\), and the complete abelian part of the Fourier expansion of the same for \(d = 7\).