Fourier Coefficients of Siegel-Eisenstein Series of Degree $2m$ and Weight $m+1$

We study Fourier coefficients of the Siegel-Eisenstein series $E_{m+1}^{(2m)}(Z)$ for $m\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $m\ge 5$. Using the Fourier expansion formula due to Mizumoto, we determine the constant term and the exceptional non-zero Fourier coefficients. The result gives a higher-degree analogue of the degree-two formulas of Kohnen and Nagaoka, which were later rederived by Haruki.