We show that, based on Grabner's recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth r that have all Fourier coefficients integral for each of r = 1, 2, 3, 4, and partly classifies them, where the classification is complete for r = 2, 3, 4; in fact, we show that there exists no normalized extremal quasimodular forms of depth 4 with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.Theorem 1.1. For r = 1, 2, 3, 4, let E r be the set of weights w such that the normalized extremal quasimodular form f (r) w of weight w and depth r has integral q-expansion. Then

We show that, based on Grabner's recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth r that have all Fourier coefficients integral for each of r = 1, 2, 3 and 4, and partly classifies them, where the classification is complete for r = 2, 3 and 4. In fact, we show that there exists no normalized extremal quasimodular forms of depth four with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.