  We introduce the family of stable Klingen congruence subgroups of GSp(4). We
use these subgroups to study both local paramodular vectors and Siegel modular
forms of degree $2$ with paramodular level. In the first part, when $F$ is a
nonarchimedean local field of characteristic zero and $(\pi,V)$ is an
irreducible, admissible representation of GSp(4,F) with trivial central
character, we establish a basic connection between the subspaces $V_s(n)$ of
$V$ fixed by the stable Klingen congruence subgroups and the spaces of
paramodular vectors in $V$ and derive a fundamental partition of the set of
paramodular representations into two classes. We determine the spaces $V_s(n)$
for all $(\pi,V)$ and $n$. We relate the stable Klingen vectors in $V$ to the
two paramodular Hecke eigenvalues of $\pi$ by introducing two stable Klingen
Hecke operators and one level lowering operator. In contrast to the paramodular
case, these three new operators are given by simple upper block formulas. We
prove further results about stable Klingen vectors in $V$ especially when $\pi$
is generic. In the second part we apply these local results to a Siegel modular
newform $F$ of degree $2$ with paramodular level $N$ that is an eigenform of
the two paramodular Hecke operators at all primes $p$. We present new formulas
relating the Hecke eigenvalues of $F$ at $p$ to the Fourier coefficients $a(S)$
of $F$ for $p^2 \mid N$. We verify that these formulas hold for a large family
of examples and indicate how to use our formulas to generally compute Hecke
eigenvalues at $p$ from Fourier coefficients of $F$ for $p^2 \mid N$. Finally,
for $p^2 \mid N$ we express the formal power series in $p^{-s}$ with
coefficients given by the radial Fourier coefficients $a(p^t S)$, $t\geq 0$, as
an explicit rational function in $p^{-s}$ with denominator $L_p(s,F)^{-1}$,
where $L_p(s,F)$ is the spin $L$-factor of $F$ at $p$.
