Journal of the
Korean Mathematical Society JKMS

In this paper, by using the modular forms of weight $nk$ ($2\leq n\in \mathbb{N}$ and $k\in \mathbb{Z}$), we construct a formula which generates modular forms of weight $2nk+4$. This formula consist of some known results in \cite{Silverman} and \cite{Aygunes-Simsek-Srivastava}. Moreover, we obtain Fourier expansion of these modular forms. We also give some properties of an operator related to the derivative formula. Finally, by using the function $j_{4}$, we obtain the Fourier coefficients of modular forms with weight $4$.

Keywords: Eisenstein series, modular forms, cusp forms, Fourier series, operators, derivative formula, theta function, Jacobi theta function

MSC numbers: Primary 11F11, 11F25; Secondary 11M36, 14K25