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We construct an explicit family of modular iterated integrals which involvescusp forms. This leads to a new method of producing "invariant versions" ofiterated integrals of modular forms. The construction will be based on anextension of higher-order modular forms which, in contrast to the standardhigher-order forms, applies to general Fuchsian groups of the first kind and,as such, is of independent interest.

Modular iterated integrals associated with cusp forms