f[x_] := Which[-1 <= x < 1, E^(-x)]
T = 2;
a[n_] := (1/T)*Integrate[f[x]*E^(-I*n*2*Pi/T*x), {x, -1, 1}];
ListPlot[Table[{k, Abs[a[k]]}, {k, -10, 10, 1}], PlotRange -> All, 
 Filling -> Axis]
ListPlot[Table[{k, Arg[a[k]]}, {k, -10, 10, 1}], PlotRange -> All, 
 Filling -> Axis]