  Unlike the $\mathcal{R}^4$ and $\nabla^4\mathcal{R}^4$ couplings, whose
coefficients are Langlands-Eisenstein series of the U-duality group, the
coefficient $\mathcal{E}_{(0,1)}^{(d)}$ of the $\nabla^6\mathcal{R}^4$
interaction in the low-energy effective action of type II strings compactified
on a torus $T^d$ belongs to a more general class of automorphic functions,
which satisfy Poisson rather than Laplace-type equations. In earlier work, it
was proposed that the exact coefficient is given by a two-loop integral in
exceptional field theory, with the full spectrum of mutually 1/2-BPS states
running in the loops, up to the addition of a particular Langlands-Eisenstein
series. Here we compute the weak coupling and large radius expansions of these
automorphic functions for any $d$. We find perfect agreement with perturbative
string theory up to genus three, along with non-perturbative corrections which
have the expected form for 1/8-BPS instantons and bound states of 1/2-BPS
instantons and anti-instantons. The additional Langlands-Eisenstein series
arises from a subtle cancellation between the two-loop amplitude with 1/4-BPS
states running in the loops, and the three-loop amplitude with mutually 1/2-BPS
states in the loops. For $d=4$, the result is shown to coincide with an
alternative proposal in terms of a covariantised genus-two string amplitude,
due to interesting identities between the Kawazumi-Zhang invariant of genus-two
curves and its tropical limit, and between double lattice sums for the particle
and string multiplets, which may be of independent mathematical interest.
