The Eisenstein ideal at prime-square level has constant rank

Let $N$ and $p$ be prime numbers with $p\ge 5$ such that $p\mathrm{\mid }\mathrm{\mid }(N+1)$. In a previous paper, we showed that there is a cuspform $f$ of weight 2 and level ${\mathrm{\Gamma }}_0(N^2)$ whose $\mathrm{\ell }$-th Fourier coefficient is congruent to $\mathrm{\ell }+1$ modulo a prime above $p$ for all primes $\mathrm{\ell }$. In this paper, we prove that this form $f$ is unique up to Galois conjugacy, and the extension of ${\mathbb{Z}}_p$ generated by the coefficients of $f$ is exactly ${\mathbb{Z}}_p[{\zeta }_p+{\zeta }_p^{-1}]$. We also prove similar results when a higher power of $p$ divides $N+1$.