Birch–Mellin formula for primitive twists

For all natural numbers $N,k,m$ with $N>0$, $k\ge 2$, and $m\ne 0$, every unital ring homomorphism $\iota:\overline{\mathbb Q}\to\mathbb C$ from the chosen algebraic closure of $\mathbb Q$, and every datum consisting of a cusp form $f$ of weight $k$ for $\Gamma_1(N)$ (viewed inside $\mathrm{GL}_2(\mathbb R)$), a Dirichlet character $\varepsilon$ modulo $N$ with values in $\overline{\mathbb Q}$, and coefficients $c_n\in\overline{\mathbb Q}$ for all natural $n$, assume that the coefficient of degree $n$ in the width-one $q$-expansion of $f$ equals $\iota(c_n)$ for every $n$, that $c_1=1$, that $f(\gamma z)=\iota(\varepsilon(d))(cz+d)^k f(z)$ for every $\gamma=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\Gamma_0(N)$ and every $z$ in the upper half-plane, and that for every prime $\ell$ and every such $z$ one has $\ell^{-1}\sum_{b=0}^{\ell-1}f((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{k-1}f(\ell z)=\iota(c_\ell)f(z)$. For every primitive $\overline{\mathbb Q}$-valued Dirichlet character $\chi$ modulo $m$ and every natural number $j$ with $j\le k-2$, define $G(\psi)=\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\psi(a))\exp(2\pi i\widetilde a/m)$, where $\widetilde a\in\{0,\ldots,m-1\}$ is the least nonnegative representative of $a$ and $\psi$ is a character modulo $m$. The asserted equality is $\displaystyle \frac{(2\pi)^{j+1}}{j!}\int_0^\infty\left(G(\chi)^{-1}\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))f(it+\widetilde a/m)\right)t^j\,dt=\frac{(-2\pi i)^jG(\chi^{-1})}{j!m^{j+1}}\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))\left(2\pi\int_0^\infty f(-\widetilde a/m+it)\bigl(m(-\widetilde a/m+it)+\widetilde a\bigr)^j\,dt\right)$. Here characters are evaluated on residue classes, $\chi^{-1}$ is the inverse Dirichlet character (with its usual zero values on nonunits), and every integral is the complex Bochner integral over the open positive real half-line with Lebesgue measure. These are total integral operations, yielding zero when the corresponding integrand is not integrable; the statement includes no separate integrability hypothesis. Complex inverses are also total, so $G(\chi)^{-1}$ means zero if $G(\chi)=0$; no separate nonvanishing hypothesis for the Gauss sum is included. The quantifiers include $m=1$, $j=0$, and $k=2$ (in the last case necessarily $j=0$), while $N=0$ and $m=0$ are excluded; since $k\ge2$, the natural-number subtraction $k-2$ is ordinary subtraction here. There is no coprimality condition between $N$ and $m$.