The bases of \(M_4(\Gamma_0(71))\), \(M_6(\Gamma_0(71))\) and the number of representation of integers (Q474519)

Summary: Following a fundamental theorem of Hecke, some bases of \(S_4(\Gamma_0(71))\) and \(S_6(\Gamma_0(71))\) are determined, and explicit formulas are obtained for the number of representations of positive integers by all possible direct sums (111 different combinations) of seven quadratic forms from the class group of equivalence classes of quadratic forms with discriminant \(-71\) whose representatives are \(x_1^2+x_1x_2+18x_2^2\), \(2x_1^2\pm x_1x_2+9x_2^2\), \(3x_1^2\pm x_1x_2+6x_2^2\) and \(4x_1^2\pm x_1x_2+5x_2^2\).