  Eta quotients on $\Gamma_0(6)$ yield evaluations of sunrise integrals at 2,
3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of
inhomogeneous differential equations whose solutions are readily obtained by
expanding in the nome $q$. Atkin-Lehner transformations that permute cusps
ensure fast convergence for all external momenta. At 4 and 6 loops, on-shell
integrals are periods of modular forms of weights 4 and 6 given by Eichler
integrals of eta quotients. Weakly holomorphic eta quotients determine
quasi-periods. A Rademacher sum formula is given for Fourier coefficients of an
eta quotient that is a Hauptmodul for $\Gamma_0(6)$ and its generalization is
found for all levels with genus 0, namely for $N = 1,2, 3, 4, 5, 6, 7, 8, 9,
10, 12, 13, 16, 18, 25$. There are elliptic obstructions at $N = 11, 14, 15,
17, 19, 20, 21, 24, 27, 32, 36, 49,$ with genus 1. We surmount these, finding
explicit formulas for Fourier coefficients of eta quotients in thousands of
cases. We show how to handle the levels $N=22, 23, 26, 28, 29, 31, 37, 50$,
with genus 2, and the levels $N=30,33,34,35,39,40,41,43,45,48,64$, with genus
3. We also solve examples with genera $4,5,6,7,8,13$.
