  We apply the methods of Fukaya, Kato and Sharifi to refine Mazur's study of
the Eisenstein ideal. Given prime numbers $N$ and $p\geq 5$ such that $p\mid
\varphi(N)$, we study the quotient of the cohomology group of modular curve
$X_{0}(N)$ by the square of the Eisenstein ideal. We study two invariants $b,c$
attached to this quotient and compute $c$. We propose a conjecture about the
invariant $b$ which relates the structure of the ray class group of conductor
$N$ to the modular symbols of $X_{0}(N)$. Assuming this conjecture, We compute
the invariant $b$.
