The function name Exp raised to a power does not make much sense. Use the base of the natural logarithm instead:
dataur=Table[7 Sqrt[15] E^(-(1/98) I (59509 t-14 x)) Tan[1/2 Sqrt[15] (-3-(59517 t)/98+3 x)],{t,-50,50,0.5},{x,-50,50,0.5}];
ListPlot3D[Abs@dataur,
Mesh->4,InterpolationOrder->2,ColorFunction->"Rainbow",PlotLabel->"q(x,t)",
PlotRange->Automatic,DataRange->{{-50,50},{-50,50}},AxesLabel->{"t","x",None}
]
