Journal of the
Korean Mathematical Society JKMS

Let $Q$ be an integral positive definite quadratic form of level $N$ in $2k(\geq4)$ variables. We assume that $(-1)^kN$ is a fundamental discriminant and the associated character $\chi$ of $Q$ is primitive of conductor $N$. Under our assumption, we find the pairs $(k,N)$ such that the dimension of spaces of cusp forms of weight $k$ and level $N$ with Nebentypus $\chi$ is one. Furthermore, we explicitly construct their bases by using Eisenstein series of lower weights. For the above pairs $(k,N)$, we use these cusp forms to provide closed formulas for the representation numbers by quadratic forms of level $N$ in $2k$ variables, which are expressed in terms of divisor functions and their convolution sums.

Keywords: Representation numbers by quadratic forms, Eisenstein series, cusp forms

MSC numbers: Primary 11E25, 11E45; Secondary 11F11

Supported by: This work was supported by the Dongguk University Research Fund of 2023 and the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2020R1F1A1A01070647).