The second moment of Dirichlet twists of Hecke $L$-functions
  Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average behavior of the square of the central value of this $L$-function? Stefanicki proved an asymptotic valid only for $q$ having very few prime factors, and we extend this to almost all $q$. 
Peng Gao, Rizwanur Khan, Guillaume Ricotta