Sums of two integer squares in arithmetic progressions

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.

• Things are not so simple in general. Have a look at the book amazon.com/Primes-Form-x2-ny2-Multiplication/dp/1118390180 – GH from MO Feb 1 at 20:14
• yes that is a great but how is it useful? the square coefficients make all the quadratic forms in the question equivalent to $x^2+y^2$ so not sure how quadratic fields etc.etc. are relevant. – Captain Darling Feb 7 at 8:33
• I believe that the kind of formula you expect does not exist. Unfortuately, I am no expert here, and I don't have the time to check the literature. – GH from MO Feb 7 at 18:50