Scalar-valued depth two Eichler-Shimura integrals of cusp forms

Given cusp forms f and g of integral weight k >= 2, the depth two holomorphic iterated Eichler-Shimura integral I-f,I-g is defined by integral(i infinity)(tau)f(z)(X-z)I-k-2(g)(z;Y)dz, where I-g is the Eichler integral of g and X,Y are formal variables. We provide an explicit vector-valued modular form whose top components are given by (If,g). We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by E-f,E-g, that can be seen as a higher-depth generalization of the scalar-valued Eichler integral E-f of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Pasol-Popa. We show that E-f,E-g can be expressed in terms of sums of products of components of vector-valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form Delta. This allows for effective computation of E-f,E-g.