ERRATUM: Peter Humphries pointed out to us that the claimed bound (3.8) (or equation (3.4) in the website version)
on Kloosterman sums can fail in the case where the conductor of th Nebentypus character \chi is not squarefree. See Example 9.3 of Knightly and Li, "Kuznetsov's trace formula and the Hecke eigenvalues of Maass forms."
However, this issue can be avoided (for the purpose of proving the main theorem) by twisting the modular forms, as we now describe.
It is enough to consider the case of forms whose Galois image is not dihedral.
Each such form can be twisted, by a Dirichlet character,
so that its conductor does not increase, but the conductor of its Nebentypus
is squarefree away from 2,3,5 and the valuation of that conductor at 2,3,5 is bounded.
This follows from results of Serre ("Modular forms of weight one and Galois representations")
an allows us to deduce the general case from the case of squarefree Nebentypus conductor (away from 2,3,5, which only affect the constant in the argument).