The construction of the bases of some eight level cusp form spaces (Q2709771)

The author wants to get formulas for the number of representations of integers by the quadratic forms NEWLINE\[NEWLINEf_{(s,k)}= 2\sum_{j=1}^k x_j^2+ \sum_{j=k+1}^s x_j^2NEWLINE\]NEWLINE with \(s\equiv 0\bmod 2\) and \(s\geq 6\) if \(k\equiv 1(2)\) and \(s\geq 10\) if \(k\equiv 0(2)\). To reach this goal, bases of the spaces of the corresponding cusp forms are constructed by theta functions with characteristics [see the second author, ibid. 4, 385-400 (1997; Zbl 0879.11017)]. The case \(s=10\) was considered by the authors in [New Trends Prob. Stat. 4, 45-55 (1997; Zbl 0924.11028)].