  Given a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a
compact Riemannian manifold and $\nabla$ a covariant derivative on $V$, let $P
= -(\lvert g \rvert^{-1/2} \nabla_\mu \lvert g \rvert^{1/2} g^{\mu\nu} u
\nabla_\nu + p^\mu \nabla_\mu +q)$ be a nonminimal Laplace type operator acting
on smooth sections of $V$ where $u,\,p^\nu,\,q$ are $M_N(\mathbb{C})$-valued
functions with $u$ positive and invertible. For any $a \in
\Gamma(\text{End}(V))$, we consider the asymptotics $\text{Tr} \,a \,e^{-tP}
\sim_{t \downarrow 0} \,\sum_{r=0}^\infty a_r(a, P)\,t^{(r-d)/2}$ where the
coefficients $a_r(a, P)$ can be written as an integral of the functions $a_r(a,
P)(x) = \text{tr}\,[a(x) \,\mathcal{R}_r(x)]$.
  This paper revisits the previous computation of $\mathcal{R}_2$ by the
authors and is mainly devoted to a computation of $\mathcal{R}_4$. The results
are presented with $u$-dependent operators which are universal (\textsl{i.e.}
$P$-independent) and which act on tensor products of $u$, $p^\mu$, $q$ and
their derivatives via (also universal) spectral functions which are fully
described.
