Let $\psi$ be an odd Dirichlet character of $G_{\mathbb{Q}}$ with conductor equal to $N$ and $p \nmid N$ be a prime number. Assume that $\psi(Frob_p)=1$.

Denote by $E_{\psi,1} \in S_1(\Gamma_1(N))$ the weight one Eisenstein series associated to the characters $(\psi,1)$ and let $f \in S_1(\Gamma_1(N)\cap \Gamma_0(p))$ be a $p$-stablization of $E_{(\psi,1)}$ (i.e $f(z)=E_{(\psi,1)}(z)-E_{(\psi,1)}(pz))$. Since $\psi(p)=1$, the evaluation of $f$ at the cusp $\infty \in X^{rig}_1(N)$ corresponding to $\mathbb{G}_{m}$ is trivial (we can see $f$ as an overconvergent modular form on $X^{rig}_1(N)$ and $\infty$ is in the ordinary locus of $X_1^{rig}(N)$).

Can we compute the evaluation of $f$ on the cusps of $X(\Gamma_1(N) \cap \Gamma_0(p))$ which are in the $\Gamma_0(p)$-orbit of $\infty$?