Fix a CM-field $K$ of degree $2g$. Let $A$ be a polarized abelian variety of dimension $n$ over $\mathbb{C}$, with an isomorphism $\theta : K \to End_{\mathbb{C}}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$. (So $n$ is a multiple of $g$.)

Then, the tangent space to the identity of $A$ defines an $n$-dimensional complex representation $\Phi$ of $K$ (which Shimura calls the type of $(A, \theta)$). 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. Then (as shown by Shimura), the representation $\Phi$ decomposes as a direct sum

$\Phi = \bigoplus_{\nu = 1}^g (r_\nu \cdot \tau_\nu \oplus s_\nu \cdot \rho \tau_\nu)$ where $r_1 + s_1 = r_2 + s_2 = \cdots = r_g + s_g = \frac{n}{g}$.

What are the possible values of $(r_1, s_1, r_2, s_2, \ldots, r_g, s_g)$ as $A$ ranges over all complex abelian varieties, while $K$ remains fixed? Can they be arbitrary nonnegative integers satisfying the above constraint, or does fixing $K$ impose further conditions?