Twisted second moment of modular $L$-functions to a fixed modulus

We study asymptotically the twisted second moment of the family of modular $L$-functions to a fixed modulus. As an application, we establish sharp lower bounds for all real $k\ge 0$ and sharp upper bounds for $k$ in the range $0\le k\le 1$ for the $2k$-th moment of these $L$-functions on the critical line.