On the number of representations of positive integers by quadratic forms as the basis of the space \(S_{4}(\Gamma_{0}(47),1)\) (Q1774686)

Summary: The number of representations of positive integers by quadratic forms \(F_{1}=x_{1}^{2}+x_{1}x_{2}+12x_{2}^{2}\) and \(G_{1}=3x_{1}^{2}+x_{1}x_{2}+4x_{2}^{2}\) of discriminant \(-47\) are given. Moreover, a basis for the space \(S_{4}(\Gamma_{0}(47),1)\) are constructed, and the formulas for \(r(n;F_{4})\), \(r(n;G_{4})\), \(r(n;F_{3} \oplus G_{1})\), \(r(n;F_{2}\oplus G_{2})\), and \(r(n;F_{1}\oplus G_{3})\) are derived.