Character twist by rational translations of a cusp form ($f_{\chi }(z)={\sum }_{j\in (\mathbb{Z}/N\mathbb{Z}{)}^{\times }}\chi (j{)}^{-1}f(z+j/N)$)

For a level-one cusp form and any Dirichlet character modulo $N>1$, this translation sum is a cusp form on ${\Gamma }_1(N)\cap {\Gamma }_0(N^2)$. Conjugating that subgroup by $\left(\begin{array}{cc}1& j/N\\ 0& 1\end{array}\right)$ yields integral determinant-one matrices; cusp holomorphy under rational slash operators supplies all cusp conditions. Its exact Fourier coefficients at positive indices are $a_n(f){\sum }_{j\in (\mathbb{Z}/N\mathbb{Z}{)}^{\times }}\chi (j{)}^{-1}{e}^{2\pi inj/N}$. For a primitive Dirichlet character they equal $g(\overline{\chi })\chi (n)a_n(f)$, by the finite Fourier transform of a primitive Dirichlet character. For imprimitive characters the sum can be nonzero at nonunits, so the simplified twist formula need not hold.

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 $U_u=\left(\begin{array}{cc}1& u/D\\ 0& 1\end{array}\right)$. If $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\Gamma }_0(N{D}^2)$, take $v\equiv d^{2}u(\mathrm{mod}D)$. 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.

Use the determinant-normalized slash operator and $W_M=\left(\begin{array}{cc}0& -1\\ M& 0\end{array}\right)$. For units $u,v$ with $Nuv\equiv -1(\mathrm{mod}D)$, the matrix identity $U_u{W}_{N{D}^2}=D{\gamma }_u{W}_N{U}_v$ has ${\gamma }_u=\left(\begin{array}{cc}(1+Nuv)/D& u\\ Nv& D\end{array}\right)\in {\Gamma }_0(N)$. A positive scalar matrix acts trivially. The nebentypus supplies $\psi (D)$; reindexing gives $\overline{\chi }(u)=\chi (-N)\chi (v)$. Thus the factor is initially $\psi (D)\chi (-N)\tau (\chi )/\tau (\overline{\chi })$. The primitive Gauss sum of a Dirichlet character identity $\tau (\chi )\tau (\overline{\chi })=\chi (-1)D$ gives the displayed equivalent factor. No assumption that $f$ itself is a Fricke eigenform is needed.