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Let $F$ be a local field, let ${\mathcal O}$ be its integer ring and $\varpi$a uniformizer of its maximal ideal. To an irreducible complex finitedimensional smooth representation $\pi$ of $GL(2,{\mathcal O})$ is associated apair of positive integers $k, k'$ called the level and the sublevel of $\pi.$The level is the smallest integer $k$ such that $\pi$ factorizes through thefinite group $GL(2,{\mathcal O}/\varpi^k {\mathcal O})$, whereas the sublevelis the smallest integer $k'\leq k$ such that there exists $\chi,$ onedimensional representation of $GL(2,{\mathcal O}),$ such that $\pi\otimes \chi$factorizes through the finite group $GL(2,{\mathcal O}/\varpi^{k'} {\mathcalO}).$ A representation of $GL(2,{\mathcal O})$ is said strongly primitive ifthe level and sublevel are equal. The classification of smooth finitedimensional representations of $GL(2,{\mathcal O})$ is equivalent to theclassification of strongly primitive irreducible representations of$GL(2,{\mathcal O}).$ In this first article we describe explicitely the evenlevel strongly primitive irreducible finite dimensional complex representationsof $GL(2,{\mathcal O})$ along the lines of A.Stasinski and R.Barrington-Leigh,G.Cliff and Q.Wen using Clifford theory. In the case where the characteristic$p$ of the residue field is not equal to $2,$ we give exact formulas for thecharacters of these representations in most cases by reducing them to theevaluation of Gauss sums, Kloosterman sums and Sali\'e sums for the finite ring${\mathcal O}/\varpi^k{\mathcal O}.$ It generalizes the work ofR.Barrington-Leigh, G.Cliff and Q.Wen which was devoted to $F={\mathbb Q}_p.$ Asecond article will give the evaluation of characters in the odd level case andthe exact expressions for certain generalized Zeta function representations of$PGL(2,{\mathcal O})$.

Evaluation of characters of smooth representations of $GL(2,\mathcal {O})$: I. Strongly primitive representations of even level