Constructions of modular forms by means of transformation formulas for theta series

TSUYUMINEFor example, this is the case where (i) $\Gamma^{\prime}=\Gamma(4)$ if $n>1$ , (ii) $\Gamma^{\prime}=\Gamma(2N^{2})$ for $N>1$ if $n\equiv 0(2)$ , (iii) $\Gamma^{\prime}=Sp_{n}(Z)$ if $n=24$ (cf.H. Maass [5]), (iv) $\Gamma^{\prime}=\Gamma(N)$ for $N\geq 2$ ifWe denote by $Z_{+},$ $Z,$ $Q,$ $R$ and $C$ , the set of all positive rational integers, the ring of rational integers, the rational number field, the real number field and the complex number field.Let $K$ be a subset of $C$ .We denote by $M_{m,n}(K)$ the set of all $m\times n$ matrices with entries in $K$ ; simply $K^{m}$ denotes $M_{m.1}(K)$and $SM_{m}(K)$ denotes the set of all symmetric matrices of degree $m$ with entries in $K$ We denote by $1_{n}$ the identity matrix of degreeWe denote the modular group $sp_{n}(Z)$ simply by $\Gamma$ .$\Gamma$ acts on the Siegel space $H_{n}$ by the usual modular transformationsHere the factor of automorphy $|CZ+D|^{1/2}$ is always determined by the conditionbe a positive integer.Then we set $\Gamma_{0}(N)=\{M\in\Gamma|C\equiv 0(N)\},$ $\Gamma(N)=\{M\in\Gamma|A\equiv D\equiv 1_{n}(N), B\equiv C\equiv 0(N)\}$ and $\Theta_{0}(N)=$ $\{M\in\Gamma_{0}(N)|({}^{t}BD)_{\Delta}\equiv 1/N({}^{t}AC)_{\Delta}\equiv(B^{l}A)_{\Delta}\equiv 1/N(D^{l}C)_{\Delta}\equiv 0(2)\}$ .For two positive integers $N_{1},$ $N_{2}$ we put $\Gamma_{0}(N_{1}, N_{2})=\{M\in\Gamma|B\equiv 0(N_{1}), C\equiv 0(N_{2})\}$ .For a positive even integer$N$ we put $\Gamma(N, 2N)=\{M\in\Gamma(N)|({}^{t}AC)_{\Delta}\equiv({}^{t}BD)_{\Delta}\equiv 0(2N),$ $\Theta_{1}(N)=\{M\in\Gamma_{0}(N)|1/N({}^{t}AC)_{\Delta}$ $\equiv 1/N(D^{t}C)_{\Delta}\equiv 0(2)\}$ and $\Theta_{2}(N)=\{M\in\Gamma_{0}(N)|({}^{t}BD)_{\Delta}\equiv(B^{l}A)_{\Delta}\equiv 0(2)\}$ .