Explicit formulae for linear characters of $\Gamma_0(N)$

We give explicit formulae for a class of complex linear unitary characters of the congruence subgroups $\Gamma_0(N)$ which involve a variant of Rademacher's $\Psi$ function. We then prove that these characters cover all characters of $\Gamma_0(N)$ precisely when $N=1,2,3,4,5,6,7,8,10,12,13$.