Consider the formal power series $${}_2F_1(1/12,5/12;1;z)^{24}-{}_2F_1(1/12,7/12;1;z)^{24}=-z/3-z^2/2-(320293/559872)z^3-\cdots$$ It follows from a theorem on modular forms that for $p\ge5$ the coefficient of $z^{p-1}$ is divisible by $p$ (more is true when considering the coefficient of $z^{p^r-1}$). Can this be proved directly without appealing to mfs ? (just out of curiosity).
Same question for $${}_2F_1(1/12,7/12;1,z)^{18}-{}_2F_1(1/6,5/6;1,z)^{18}$$
ADDED:
I realize that I should have asked first much simpler similar questions: ${}_2F_1(1/12,5/12;1,z)^k$ for $k=1,2,3,4,5,6,8,10$, and ${}_2F_1(1/12,7/12;1,z)^k$ for $k=1,2,3,4,6,7,9$.
