Large Values of Newforms on GL(2) with Highly Ramified Central Character

We give a lower bound for the sup-norm of an |L²|-normalized newform in an irreducible, unitary, cuspidal representation |π| of |GL₂| over a number field. When the central character of |π| 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 |π|⁠. 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.