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.

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    $\begingroup$ Things are not so simple in general. Have a look at the book amazon.com/Primes-Form-x2-ny2-Multiplication/dp/1118390180 $\endgroup$ – GH from MO Feb 1 at 20:14
  • $\begingroup$ 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. $\endgroup$ – Captain Darling Feb 7 at 8:33
  • $\begingroup$ 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. $\endgroup$ – GH from MO Feb 7 at 18:50

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