Belyi functions of the special weighted trees of $(2,3)$-type

In this paper we calculate the Belyi functions of the weighted trees of $(2,3)$-type with primitive special edge rotation groups. There are 21 Galois orbits of these trees: 6 rational orbits, 12 orbits over quadratic fields, and three orbits over fields of degree 3, 4 and 6. The highest degree of the calculated Belyi function is 32. The calculations are performed using modular functions techniques.