Construction of Generalized Harish-Chandra Modules with Arbitrary Minimal $\mathfrak k$-Type Let $\mathfrak g$ be a semisimple complex Lie algebra and $\k\subset\g$ be any algebraic subalgebra reductive in $\mathfrak g$. For any simple finite dimensional $\mathfrak k$-module $V$, we construct simple $(\mathfrak g,\mathfrak k)$-modules $M$ with finite dimensional $\mathfrak k$-isotypic components such that $V$ is a $\mathfrak k$-submodule of $M$ and the Vogan norm of any simple $\k$-submodule $V'\subset M, V'\not\simeq V$, is greater than the Vogan norm of $V$. The $(\mathfrak g,\mathfrak k)$-modules $M$ are subquotients of the fundamental series of $(\mathfrak g,\mathfrak k)$-modules. Categories:17B10, 17B55