I think this is an easy question, but I need some time to introduce it. I need to apply Yumiko Hironaka's computations on local densities of hermitian forms (see [1]).
I would have liked to create the new tag [local-densities], but I don't have enough reputation. 

Let $\mathcal O$ be a ring of integers of an imaginary quadratic number field, like $\mathbb Z(\sqrt{-1})$. 
Let $H\in \mathrm{GL}(n,\mathcal O)$ be a Hermitian matrix and $\ell$ an integer number.
For $p$ an integer prime number, I need to compute the $p$-local density given by
$$
\delta_p(H,\ell) = \lim_{j\to\infty} \frac{A_j(H,\ell)}{p^{j(2n-1)}}
$$
where 
$$
A_j(H,\ell) = \\# \{ x\in ({{\mathcal O}/p^j{\mathcal O}})^n:x^*Hx\equiv \ell\pmod {p^j} \}.
$$



Now I introduce the setting on Hironaka's paper: let $k$ be a nonarchimedian local field of characteristic $0$, $\mathcal O_k$ the ring of integers in $k$ (note that $\mathcal O$ and $\mathcal O_k$ are different rings), $*$ an involution on $k$ and denote $k_0$ the fixed field by $*$. Assume that $k$ is unramified over $k_0$. Let $q$ be the residue class field of $k_0$, $\pi\in k_0$ be a prime element of $k$ and $\mathfrak{p}=\pi\mathcal O_k$.
She gives a formula for 
$$
\mu_p(H,\ell) = \lim_{j\to\infty} \frac{N_j(H,\ell)}{q^{j(2n-1)}}
$$
where 
$$
N_j(H,\ell)= \\# \{ x\in ({{\mathcal O_k}/\mathfrak{p}})^n:x^*Hx\equiv \ell\pmod {\mathfrak{p}^j} \}.
$$

Let $D_{\mathcal O}(<0)$ be the discriminant of $\mathcal O$. 


If $(\frac{D_{\mathcal O}}{p})=-1$, we have that $p\mathcal O$ is a prime ideal in $\mathcal O$, and we can apply Hironaka's formula to the nonarquimedian local field $k=\mathbb Q[\sqrt{D_\mathcal O}]\otimes_\mathbb Q \mathbb Q_p$, where the involution $*$ is given by $\alpha\otimes x\mapsto \bar\alpha\otimes x$, and $k_0=\mathbb Q_p$. 
This can be done because $\mathcal O_k/\mathfrak{p} \simeq \mathcal O/p\mathcal O$, since $\mathfrak{p}=p\mathcal O_k$.


Now, if $(\frac{D_{\mathcal O}}{p})=+1$, we have that $p\mathcal O$ decomposes as a product of two different prime ideals (are conjugated). Also, $\mathbb Q[\sqrt{D_\mathcal O}]\otimes_\mathbb Q \mathbb Q_p$ is isomorphic to $\mathbb Q_p\times \mathbb Q_p$. 
Finally, my question is: 

How do I apply Hironaka's formula to this case? Who is the local field $k$?
Who is the conjungation on $k$ and the fixed field $k_0$? 
Who is the prime element $\pi$?


I'm sorry for this long question. I hope that somebody helps me with an understandable answer since I usually don't work in this area. Thanks.-.



[1] Y. Hironaka. "Local zeta functions on hermitian forms and its application to local densities". Journal of Number Theory 71, 40--64 (1998). [Link][1]. 


  [1]: http://www.sciencedirect.com/science/article/pii/S0022314X98922375