  In this paper we apply results from the theory of congruences of modular
forms (control of reducible primes, level-lowering), the modularity of elliptic
curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to
show the existence of newforms of weight 2 and trivial nebentypus with
coefficient fields of arbitrarily large degree and square-free or almost
square-free level. More precisely, we prove that for any given numbers t and B,
there exists a newform f of weight 2 and trivial nebentypus whose level N is
square-free (almost square-free), N has exactly t prime divisors (t odd prime
divisors and a small power of 2 dividing it, respectively), and the degree of
the field of coefficients of f is greater than B.
