Boundary Hecke sum at a single cusp

Let $N>0$, let $\Phi$ be a $\Gamma_1(N)$-equivariant homogeneous polynomial-valued function of degree $n$ on the rational cusps, and let $\ell$ be a prime with $\ell\equiv1\pmod N$. For the standard $\ell+1$ representatives $\begin{pmatrix}1&b\\0&\ell\end{pmatrix}$ ($b\bmod\ell$) and $\begin{pmatrix}\ell&0\\0&1\end{pmatrix}$, the sum of the adjugate-normalized transforms of $\Phi$ at any cusp $x$ is $(1+\ell^{n+1})\Phi(x)$. This is the single-cusp calculation underlying the scalar action on boundary cochains. In primitive-vector terms, one representative produces a vector of content $\ell$ and the other $\ell$ representatives produce primitive vectors; homogeneity accounts for the power $\ell^{n+1}$.