  Let $k$ and $N$ be positive integers with $k\ge2$ even. In this paper we give
general explicit upper-bounds in terms of $k$ and $N$ from which all the
residual representations $\bar{\rho}_{f,\lambda}$ attached to non-CM newforms
of weight $k$ and level $\Gamma_0(N)$ with $\lambda$ of residue characteristic
greater than these bounds are "as large as possible". The results split into
different cases according to the possible types for the residual images and
each of them is illustrated on some numerical examples.
