Explicit Large Image Theorems for Modular Forms

Let $k$ and $N$ be positive integers with $k\ge 2$ 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 ${\mathrm{\Gamma }}_0(N)$ 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.