= Primitive twist at coprime level
{title2=$f_\chi\in S_k(\Gamma_0(ND^2),\psi\chi^2),\quad(N,D)=1$}
For a <primitive Dirichlet character> $\chi$ modulo $D$, its <Gauss sum of a Dirichlet character> gives $f_\chi=\tau(\overline\chi)^{-1}\sum_u\overline\chi(u)f|U_u$, with $U_u=\begin{pmatrix}1&u/D\\0&1\end{pmatrix}$. If $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma_0(ND^2)$, take $v\equiv d^2u\pmod D$. Then $U_u\gamma U_v^{-1}\in\Gamma_0(N)$ and its lower-right entry is congruent to $d$ modulo $N$. Reindexing the sum yields the character $\psi(d)\chi(d)^2$. Rational translations preserve vanishing at <modular cusps>, proving the stated cuspidality as well as the transformation law.
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