  In this paper, we propose an improved algorithm for computing mod $\ell$
Galois representations associated to eigenforms of arbitrary levels prime to
$\ell$. Precisely, we present a method to find the Jacobians of modular curves
which have the smallest possible dimensions in a well-defined sense to realize
the modular Galois representations. This algorithm also works without the
assumption $\ell \ge k-1$, where $k$ are the weights of the eigenforms.
