# Transformation property for classical Siegel modular forms of weight 2

Let $\mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\}$ be the Siegel upper half space. There are Eisenstein series $$E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap \Gamma)\backslash \Gamma}{1\big|_{2k}\gamma} = \sum_{\gamma\in (P_0\cap \Gamma)\backslash \Gamma}{det(C_\gamma \tau + D_\gamma)^{-2k}}$$ on $\mathbb{H}_g$ (here $P_0$ is the subgroup of $\Gamma:=Sp_{2g}(\mathbb{Z})$ consisting of those matrices whose lower left $g\times g$-block is the zero matrix). Consider the following normalization $$G_{2k}(\tau):= \frac{(2k-1)!\zeta(2k)}{(2\pi i)^{2k}}\cdot E_{2k}(\tau).$$ For $g=1$ one knows that $G_2(\tau)$ transforms as follows w.r.t. an element $\begin{pmatrix} a & b \\ c & d \end{pmatrix} =:\gamma \in SL_2(\mathbb{Z})$ $$G_2(\frac{a\tau + b}{c\tau +d}) (c\tau +d)^{-2} = G_2(\tau) - \frac{c}{4\pi i(c\tau + d)}.$$ Does anybody know how this transformation looks like for $g>1$ w.r.t. $\, \gamma \in Sp_{2g}(\mathbb{Z})$ and/or indicate literature on this?

I'm only interested in classical (meaning scalar-valued) Siegel modular forms.