Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to count integer representations of $n$ by the quadratic form $x^2+ 5^2 y^2$.

I would hope for something as nice as the formula related to representations as a sum of two integer squares, $$ \sum_{\substack{ d\in \mathbb{N} \\ d \text{ divides } n}} \chi(d),$$ where $\chi$ is the non-principal character modulo $4$.

In general, I would be interested in the number of representations by any quadratic form of the shape $d_1^2 x^2+ d_2^2 y^2$, where $d_1,d_2$ are non-zero integers.