  Brumer and Kramer gave bounds on local conductor exponents for an abelian
variety $A/\mathbb Q$ in terms of the dimension of $A$ and the localization
prime $p$. Here we give improved bounds in the case that $A$ has maximal real
multiplication, i.e., $A$ is isogenous to a factor of the Jacobian of a modular
curve $X_0(N)$. In many cases, these bounds are sharp. The proof relies on
showing that the rationality field of a newform for $\Gamma_0(N)$, and thus the
endomorphism algebra of $A$, contains $\mathbb Q(\zeta_{p^r})^+$ when $p$
divides $N$ to a sufficiently high power. We also deduce that certain
divisibility conditions on $N$ determine the endomorphism algebra when $A$ is
simple.

Number Theory