Fix a CM-field $K$ of degree $2g$, and a natural number $n$ which is a multiple of $g$. Write
$\tau_1, \tau_2, \ldots, \tau_g, \rho \tau_1, \rho \tau_2, \ldots, \rho \tau_g$
for the different embeddings of $K$ into $\mathbb{C}$, where $\rho$ denotes complex complex conjugation. Let $\Phi$ be an $n$-dimensional complex representation of $K$ in the form
$\Phi = \bigoplus_{\nu = 1}^g (r_\nu \cdot \tau_\nu \oplus s_\nu \cdot \rho \tau_\nu)$ where $r_1 + s_1 = r_2 + s_2 = \cdots = r_g + s_g = \frac{n}{g}$.
Consider the moduli space of pairs $(A, \theta)$ where $A$ is an $n$-dimensional abelian variety and $\theta : K \hookrightarrow End(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ is an injection such that the $n$-dimensional complex representation of $K$ defined by the tangent space to the identity of $A$ is isomorphic to $\Phi$.
In terms of $(r_1, s_1, \ldots, r_g, s_g)$, what is the minimal field over which this moduli space is defined? Moreover, what is the minimal extension of this field over which all of its irreducible components are defined?
EDIT: As pointed out by Kevin Buzzard below, what I really want is the minimal field not just where the moduli space is defined, but where the corresponding moduli functor is also defined. Moreover (as pointed out by Keerthi below), we should probably also fix a polarization of $A$, in order to guarantee that this moduli space exists.
