Quotients of theta series as rational functions of \(J\) and \(\lambda\) (Q1112091)

Let \(H=\{z\in\mathbb{C}: \Im(z)>0\}\), \(\hat {\mathbb{Q}}=\mathbb{Q}\cup \{\infty \}\) and \(\Gamma (N)=\{M\in \mathrm{SL}_ 2(\mathbb{Z}): M\equiv I \bmod N\}\) \((N=1,2,3,\ldots)\). Set \(H^*=H\cup\hat {\mathbb{Q}}\). Then the orbit space \(\bar{\Gamma}(N) \setminus H^*\) (\(\bar{\Gamma}(N) = \Gamma (N)\setminus \pm I)\) which we denote by \(X(N)\) is a compact Riemann surface with genus \(g_N\). By \(K(X(N))\) we mean the field of meromorphic functions on the compact Riemann surface \(X(N)\). Since the genus is 0 for \(N=1\) or \(2\), we have theoretically field isomorphisms \(K(X(1))\cong \mathbb{C}(J)\) and \(K(X(2))\cong\mathbb{C}(\lambda)\) where \(\mathbb{C}(J)\) and \(\mathbb{C}(\lambda)\) are the rational function fields of classical modular functions \(J\) and \(\lambda\), respectively. We find algorithms by working with the quotients of theta series \(\theta_A(\tau)\) (\(\tau\in H)\) as modular functions of level 1 and 2. The theta series \(\theta_A(\tau)\) are associated with unimodular even positive definite integral quadratic forms \(A[X]\) in \(n\) \((\equiv 0 \bmod 8)\) variables.