To find the coordinates of point \( C' \) after dilating point \( C(0,3) \) with a scale factor of 2 and a center of dilation at the origin, we can use the following formula for dilation:
\[ (x', y') = (kx, ky) \]
where \( (x, y) \) are the original coordinates, \( k \) is the scale factor, and \( (x', y') \) are the coordinates after dilation.
For point \( C(0, 3) \) and a scale factor \( k = 2 \):
\[ x' = 2 \cdot 0 = 0 \] \[ y' = 2 \cdot 3 = 6 \]
Thus, the coordinates of point \( C' \) after dilation are:
\[ C' (0, 6) \]
So, the answer as an ordered pair is:
\[ (0, 6) \]