In Miyake's book on modular forms explicit formulas for the $q$-expansions of a basis of the space Eisenstein series of arbitrary level and weight were given. I guess similar formulas for a basis of the entire space cuspidal forms do not exist. Is there at least an explicit formula for the $q$-expansion of a non-zero cuspidal holomorphic modular form of level $\Gamma_0(p)$ and weight $2$ for arbitrarily large primes $p$?