  We consider a class of differential equations for multi-loop Feynman
integrals which can be solved to all orders in dimensional regularisation in
terms of iterated integrals of meromorphic modular forms. We show that the
subgroup under which the modular forms transform can naturally be identified
with the monodromy group of a certain second-order differential operator. We
provide an explicit decomposition of the spaces of modular forms into a direct
sum of total derivatives and a basis of modular forms that cannot be written as
derivatives of other functions, thereby generalising a result by one of the
authors form the full modular group to arbitrary finite-index subgroups of
genus zero. Finally, we apply our results to the two- and three-loop equal-mass
banana integrals, and we obtain in particular for the first time complete
analytic results for the higher orders in dimensional regularisation for the
three-loop case, which involves iterated integrals of meromorphic modular
forms.
