  Consider the Fourier expansions of two elements of a given space of modular
forms. How many leading coefficients must agree in order to guarantee that the
two expansions are the same? Sturm gave an upper bound for modular forms of a
given weight and level. This was adapted by Ram Murty, Kohnen and Ghitza to the
case of two eigenforms of the same level but having potentially different
weights. We consider their expansions modulo a prime ideal, presenting a new
bound. In the process of analysing this bound, we generalise a result of Bach
and Sorenson, who provide a practical upper bound for the least prime in an
arithmetic progression.
