Large values of newforms on GL(2) with highly ramified central character

We give a lower bound for the sup-norm of an $L^2$-normalized newform in an irreducible, unitary, cuspidal representation $\pi$ of $GL_2$ over a number field. When the central character of $\pi$ is sufficiently ramified, this bound improves upon the trivial bound by a positive power of $N$ where $N$ is the norm of the conductor of $\pi$. This generalizes a result of Templier, who dealt with the special case when the conductor of the central character equals the conductor of the representation. We also make a conjecture about the true size of the sup-norm in the $N$-aspect that takes into account this central character phenomenon. Our results depend upon some explicit formulas and bounds for the Whittaker newvector over a non-archimedean local field, which may be of independent interest.