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\begin{document}$$k \ge 2$$\end{document} and N be positive integers and let χ\documentclass[12pt]{minimal}
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\begin{document}$$\chi $$\end{document} be a Dirichlet character modulo N. Let f(z) be a modular form in Mk(Γ0(N),χ)\documentclass[12pt]{minimal}
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\begin{document}$$M_k(\Gamma _0(N),\chi )$$\end{document}. Then we have a unique decomposition f(z)=Ef(z)+Sf(z)\documentclass[12pt]{minimal}
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\begin{document}$$f(z)=E_f(z)+S_f(z)$$\end{document}, where Ef(z)∈Ek(Γ0(N),χ)\documentclass[12pt]{minimal}
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\begin{document}$$E_f(z) \in E_k(\Gamma _0(N),\chi )$$\end{document} and Sf(z)∈Sk(Γ0(N),χ)\documentclass[12pt]{minimal}
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\begin{document}$$S_f(z) \in S_k(\Gamma _0(N),\chi )$$\end{document}. In this paper, we give an explicit formula for Ef(z)\documentclass[12pt]{minimal}
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\begin{document}$$E_f(z)$$\end{document} in terms of Eisenstein series whose coefficients are sum of divisors function. Then we apply our result to certain families of eta quotients and to representations of positive integers by 2k–ary positive definite quadratic forms in order to give an alternative version of Siegel’s formula for the weighted average number of representations of an integer by quadratic forms in the same genus. Our formula for the latter is in terms of sum of divisors function and does not involve computation of local densities.
