13. (15 points) As shown in the figure, there is a convex-shaped paper piece composed of four $1 \times 1$ small squares and a $5 \times 6$ grid paper. Now, the convex-shaped paper piece is to be glued onto the grid paper, with each small square of the convex-shaped paper piece overlapping with a small square on the grid paper. How many different shapes can be formed? (Two shapes that are the same after rotation are considered the same shape)

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【Analysis】We can discuss the problem by dividing it into cases, referring to the small square at the top of the convex shape as its head. The shapes formed by sticking can be divided into two categories: the first category is when the head of the convex shape is on the edge of the grid paper, and the second category is when the head of the convex shape is inside the grid paper.

【Solution】According to the analysis, we refer to the small square at the top of the convex shape as its head, and the shapes formed by sticking can be divided into two categories:

For the first category, the head of the convex shape cannot be stuck at the four corners of the grid paper. The small squares on the edges (excluding the corners) of the grid paper total: $2 \times 3 + 2 \times 4 = 14$ (squares), resulting in 14 shapes.

For the second category, each small square inside the grid paper can have the head of the convex shape stuck on it, and the head can face up, down, left, or right.

Therefore, this category has $4 \times (3 \times 4) = 48$ (shapes).
By the principle of addition, there are a total of $14 + 48 = 62$ shapes. Since each small square on the grid paper is rotationally symmetric with another small square,

the total number of different shapes is: $62 \div 2 = 31$ (shapes).
Therefore, the answer is: 31.