  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 finite
dimensional smooth representation $\pi$ of $GL(2,{\mathcal O})$ is associated a
pair 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 the
finite group $GL(2,{\mathcal O}/\varpi^k {\mathcal O})$, whereas the sublevel
is the smallest integer $k'\leq k$ such that there exists $\chi,$ one
dimensional representation of $GL(2,{\mathcal O}),$ such that $\pi\otimes \chi$
factorizes through the finite group $GL(2,{\mathcal O}/\varpi^{k'} {\mathcal
O}).$ A representation of $GL(2,{\mathcal O})$ is said strongly primitive if
the level and sublevel are equal. The classification of smooth finite
dimensional representations of $GL(2,{\mathcal O})$ is equivalent to the
classification of strongly primitive irreducible representations of
$GL(2,{\mathcal O}).$ In this first article we describe explicitely the even
level strongly primitive irreducible finite dimensional complex representations
of $GL(2,{\mathcal O})$ 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 the
characters of these representations in most cases by reducing them to the
evaluation of Gauss sums, Kloosterman sums and Sali\'e sums for the finite ring
${\mathcal O}/\varpi^k{\mathcal O}.$ It generalizes the work of
R.Barrington-Leigh, G.Cliff and Q.Wen which was devoted to $F={\mathbb Q}_p.$ A
second article will give the evaluation of characters in the odd level case and
the exact expressions for certain generalized Zeta function representations of
$PGL(2,{\mathcal O})$.
