Cusp forms in $S_{4}left( Gamma _{0}left( 79right) right) $ and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant $-79$

摘要

A basis of a subspace of $S_{4}left( Gamma _{0}left( 79ight) ight) $ is given and the formulas for the number of representations of positive integers by some direct sums of the quadratic forms $x_{1}^{2}+x_{1}x_{2}+20x_{2}^{2},$ $ 4x_{1}^{2}pm x_{1}x_{2}+5x_{2}^{2},$ $2x_{1}^{2}pm x_{1}x_{2}+10x_{2}^{2}$ are determined.