  If $\mathfrak{g}$ is a contragredient Lie superalgebra and $\gamma$ is a root
of $\mathfrak{g},$ we prove the existence and uniqueness of \v{S}apovalov
elements for $\gamma$ and give upper bounds on the degrees of their
coefficients. Then we use \v{S}apovalov elements to define some new highest
weight modules. If $X$ is a set of orthogonal isotropic roots and $\lambda \in
\mathfrak{h}^*$ is such that $\lambda +\rho$ is orthogonal to all roots in $X$,
we construct a highest weight module $M^X(\lambda)$ with character
$\epsilon^\lambda {p}_X$. Here $p_X$ is a function that counts partitions not
involving roots in $X$. Examples of such modules can be constructed via
parabolic induction provided $X$ is contained in the set of simple roots of
some Borel subalgebra. However our construction works without this condition
and provides a highest weight module for the distinguished Borel subalgebra.
The main results are analogs of the \v{S}apovalov determinant and the Jantzen
sum formula for $M^X(\lambda)$ when $\mathfrak{g}$ has type A.For the proof it
is enough to study the behavior for certain "relatively general" highest
weights. Using an equivalence of categories due to Cheng, Mazorchuk and Wang,
the information we require is deduced from the behavior of the modules
$M^X(\lambda)$ when $\mathfrak{g}=\mathfrak{gl}(2,1)$ or $\mathfrak{gl}(2,2)$.
These low dimensional cases are studied in detail in an appendix.
