Computing a series using the $p$-adic norm

Question

For my research, I am interested in studying the series $$M_{k,D}^{(p)}(x)=\sum_{M=1}^\infty\frac{1}{p^{Mk}}\sum_{\substack{a,b,c\in \mathbb{Z}\\p\nmid gcd(a,b,c)\\b^2-4ac=Dp^{2M}\\a<0<Q(x)}}(ax^2+bx+c)^k$$ If $k$ is an integer, then there is nothing to do this series diverges in the euclidean norm. If we look at the convergence with respect to the $p$-adic norm by considering $M_{k,D}^{(p)}$ as a function on $\mathbb{Q}$, then the series should converge for all $k<0$. My question is : Is there any efficient way to compute the p-adic value of this series ? Here I am interested in $D=0,1mod(4)$ not a square where $p$ is a prime number and $Q(x)=ax^2+bx+c$. When I say $Q(x)>0$ I mean for a fix $x$ I consider only those quadratics that are positive at $x$. I am not too comfortable with $p$-adic techniques so any help would be helpful !