@StanleyYaoXiao Oh, thank you. So if I take an arbitrary quadratic form of the same discriminant as $x^{2}-dy^{2}$ and find the number of representations of $N$ by both up to $GL_{2}(\mathbb{Z})$-equivalence then is it going to be the same number?

@StanleyYaoXiao Yes, you're right. Do you know, by chance, any good papers on the way this number of representations is counted? I want to have a deeper understanding.