Using a one-dimensional tight-binding Anderson model, we study a disordered nanowire in the presence of an external gate which can be used for depleting its carrier density (field effect transistor device configuration). In this first paper, we consider the low temperature coherent regime where the electron transmission through the nanowire remains elastic. In the limit where the nanowire length exceeds the electron localization length, we derive three analytical expressions for the typical value of the thermopower as a function of the gate potential, in the cases where the electron transport takes place (i) inside the impurity band of the nanowire, (ii) around its band edges and eventually (iii) outside its band. We obtain a very large enhancement of the typical thermopower at the band edges, while the sample to sample fluctuations around the typical value exhibit a sharp crossover from a Lorentzian distribution inside the impurity band towards a Gaussian distribution as the band edges are approached.