You asked

Best telephoto lens for the fujifilm x e1?

Here is what the site says about that.

From Fujifilm X-E1: Fujifilm X-E1: A mirrorless camera by Fujifilm with a 16.3-megapixel APS-C sensor, announced in 2012. Specs - Maker: Fujifilm - Mount: Fujifilm X - Sensor: APS-C, 23.5 x 15.6mm, crop factor 1.5 - Resolution: 16.3 megapixels - Stabilisation: none in the body; use stabilised lenses for slow shutter speeds - Announced: 2012 - Type: mirrorless camera Focal lengths on it Lenses frame like 1.5 times their focal length on full frame: 16mm ...

From Sensor sizes: What it costs you: - Size, weight and price, especially of the lenses. A full-frame telephoto is much larger than its APS-C or Micro Four Thirds equivalent. - Less depth of field when you want more of it, for macro or landscape.

Quoted from this site's pages.

From these pages