I am looking for a reference request for a fact I believe to be true (or a confirmation that it is wrong). We may parameterize the algebraic representations of $GL_n(\mathbb{C})$ by non-increasing sequences of $n$-integers $\kappa = \kappa_1 \geq \ldots \geq \kappa_n$, with $\mathrm{diag}(\lambda_1, \ldots \lambda_n)$ acting by $\lambda_1^{\kappa_1} \cdot \ldots \cdot \lambda_n^{\kappa_n}$, and in a slight abuse of notation we'll call this a representation $(\kappa, V_\kappa)$.
For $\Gamma \subset SP_n(\mathbb{Z})$ a finite index subgroup, say that a Siegel modular form of weight $\kappa$ and level $\Gamma$ is a holomorphic function $f: H_n \rightarrow V_\kappa$ (where $H_n$ is the Siegel upper half space of genus $n$) if it satisfies
$$ f(\gamma \tau) = \kappa( c \tau + d ) f(\tau) $$
for all $\gamma = \begin{pmatrix} a & b \\\ c & d \end{pmatrix} \in \Gamma$. I believe it to be the case that there only exist nonzero Siegel modular forms for $\kappa_n > 0$ (perhaps even better $\kappa_n > n$, but I am not sure). Is this the case, and if so do you know a good reference? If it is not true for all levels $\Gamma$, is it at least true for $\Gamma = SP_n(\mathbb{Z})$?