  Let $K$ be an imaginary quadratic field, $N$ be a positive integer, $f(z)$ be
a newform of level $\Gamma_1(N)$, and $A_f$ be the abelian variety associated
to $f$. For each $\tau \in K$ ($\operatorname{Im} \tau >0$), we construct a
certain point $P_\tau$ on $A_f$ defined over an extended ring class field of
$K$ of level $N$. Our construction generalizes Birch's construction of the
Heegner points to the abelian varieties associated to modular forms of level
$\Gamma_1(N)$ and nontrivial character. Then, we show that $P_\tau$'s satisfy
the distribution and congruence relations of an Euler system, which implies
that it should be possible to apply the Euler system techniques to them to show
a relation between the non-torsionness of $P_\tau$ and the rank of $A_f(K)$.
