Cusp forms in \(S_4(\Gamma_{0}(79))\) and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant \(-79\) (Q2890348)

In the present paper, the author gives a basis of a subspace of \(S_{4}(\Gamma _{0}(79))\) and the formulas for the number of the representations of positive integers of the quadratic forms \(x_{1}^{2}+x_{1}x_{2}+20x_{2}^{2}\), \(4x_{1}^{2}\pm x_{1}x_{2}+5x_{2}^{2}\) and \(2x_{1}^{2}\pm x_{1}x_{2}+10x_{2}^{2}\). Even though the results are true there are unfortunately a lot of mistakes in many pages of the paper. For example, the estimates are wrong and the given result can not be verified from the above calculations. The same errors are seen in the whole paper. It is surprising that the results are true even though the estimates are wrong! A list of the errors is given in the following:NEWLINENEWLINE1. On page 537 \(\downarrow 5\): \(x_{1}x_{2}-\frac{2}{79}(F_{1}\oplus \Psi _{1})\) is false, \(x_{1}x_{2}+\frac{1}{2\cdot 79}(F_{1}\oplus \Psi _{1})\) is true.NEWLINENEWLINE2. On page 545 \(\downarrow 9,10,12:\) \(q^{n}\) forgotten.NEWLINENEWLINE3. On page 551 \(\downarrow 18\): \(\frac{1}{79}(2q+16q^{4}+\cdots)\) is false, \( \frac{1}{2\cdot 79}(2q+16q^{4}+\cdots)\) is true.NEWLINENEWLINE4. On page 554 \(\downarrow 10:\) \(+(-10\cdot 5\cdot 4)q^{5}\) is false, \( +0q^{5}\) is true since it has no integer solutions for \(n=5\).NEWLINENEWLINE5. On page 554 \(\downarrow 15:\) \((79\cdot 9\cdot 2+79\cdot 4\cdot 4+79\cdot 1\cdot 4-10\cdot 18\cdot 20)q^{18}\) is false, \((79\cdot9\cdot2+79\cdot4 \cdot8+79\cdot1\cdot4-10\cdot18\cdot20)q^{18}\) is true.NEWLINENEWLINE6. On page 554 \(\downarrow 18:\) \(-666q^{18}\) is false, \(+666q^{18}\) is true.NEWLINENEWLINE7. On page 554 \(\uparrow 4:\) \((79\cdot1\cdot10-2\cdot18\cdot20)q^{18}\) is false, \((79\cdot1\cdot8-2\cdot18\cdot20)q^{18}\) is true.NEWLINENEWLINE8. On page 556 \(\downarrow 13:\) \((79\cdot1\cdot4+79\cdot9\cdot4+79\cdot4 \cdot4-20\cdot12\cdot8)q^{12}\) is false, \((79\cdot1\cdot4+79\cdot4\cdot4-20 \cdot12\cdot8)q^{12}\) is true.NEWLINENEWLINE9. On page 556 \(\downarrow 16:\) \((79\cdot1\cdot4+79\cdot9\cdot4-20\cdot17 \cdot12)q^{17}\) is false, \((79\cdot1\cdot4+79\cdot4\cdot4+79\cdot9\cdot4-20 \cdot17\cdot12)q^{17}\) is true.NEWLINENEWLINE10. On page 556 \(\uparrow 9:\) \(-120q^{19}\) is false, \(+120q^{19}\) is true.NEWLINENEWLINE11. On page 557 \(\downarrow 3:\) \((79\cdot4\cdot8+79\cdot1\cdot8-10\cdot17 \cdot12)q^{17}\) is false, \((79\cdot4\cdot8+79\cdot1\cdot4-10\cdot17 \cdot12)q^{17}\) is true.NEWLINENEWLINE12. On page 557 \(\downarrow 4:\) \((79\cdot9\cdot2+79\cdot1\cdot4-10\cdot19 \cdot6)q^{18}\) is false, \((79\cdot9\cdot2+79\cdot1\cdot4-10\cdot18 \cdot6)q^{18}\) is true.NEWLINENEWLINE13. On page 560 \(\downarrow 9:\) \((79\cdot1\cdot12+79\cdot4\cdot4-4\cdot18 \cdot18)q^{18}\) is false, \((79\cdot1\cdot12-4\cdot18\cdot18)q^{18}\) is true.NEWLINENEWLINE14. On page 562 \(\uparrow 5:\) \(+\frac{135840}{3121}q^{11}\) is false, \(+\frac{ 159840}{3121}q^{11}\) is true since \(120\cdot 1332=159840\). Otherwise all coefficients obtained in Section 6 are not correct!NEWLINENEWLINE15. On page 563 \(\downarrow 4:\) \(+\frac{163776}{3121}q^{11}\) is false, \(+ \frac{139776}{3121}q^{11}\) is true since \(96-\frac{159840}{3121}=\frac{139776 }{3121}\).NEWLINENEWLINE16. On page 567 \(\uparrow 7:\) \(100192/312\) is false, \(100192/3121\) is true.11