Can an ideal ellipsoid of rotation be espressed exactly with a spherical harmonic series of a finite number of terms?

Question

The ellipsoid of rotation has rotational symmetry, therefore the coefficient of every tesseral and every sectorial base function is zero. The ellipsoid has equatorial symmetry, therefore coefficients of all zonal base functions of odd degree are zero. More particularly, if $Y_l^m \left( \theta \right)$ represents the spherical harmonic basis function of degree $l$ and order $m$, and $a_l^m \in \, \mathbb{R}$ is the corresponding series coefficient, then does there exist the set $\left\{ a_{2i}^0 \right\} : 0 \leq i \in \mathbb{N} \leq k \in \mathbb{R}$ with $\sum_{i = 0}^{k} a_{2i}^0 Y_{2i}^0 (\theta) = E$ such that $E$ is exactly ellipsoidal?
I read somewhere, in The Geodesy Literature, that GRS80 ellipsoid is defined by the series $\sum_{i = 0}^{8} a_{i}^0 Y_{i}^0 (\theta) $ . My question is whether with max degree 8, it bears an actual symbolical equivalence to an ellipsoid, or contrarily, if degree 8 is, for example, just a conventional approximation.