Cusp forms in \(S_{6}(\varGamma_{0}(23))\), \(S_{8}(\varGamma_{0}(23))\) and the number of representations of numbers by some quadratic forms in 12 and 16 variables (Q457011)

The paper follows the work of \textit{G. Lomadze} [Georgian Math. J. 4, No. 6, 523--532 (1997; Zbl 0885.11028)]. The author constructs bases of the space of cusp forms \(S_6(\Gamma_0(23))\) and \(S_8(\Gamma_0(23))\) and uses them to get explicit formulae for the number of representations of positive integers by some quadratic forms in \(12\) and \(16\) variables that are direct sums of the binary quadratic forms \(x^2_1+ x_1x_2+ 6x^2_2\) and \(2x^2_1+ x_1x_2+ 3x^2_2\) with discriminant \(-23\).