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By results of Aoki and Ibukiyama [MR:2130626, 10.1142/S0129167X05002837] , and of Hayashida and Ibukiyama [MR:1840071] , the module $\oplus_k M_{k}(\Gamma_0(4),\psi_4)$ of Siegel modular forms of degree 2 with respect to the group $\Gamma_0(4)$ and character $\psi_4$ is generated by the following functions, which are defined in terms of theta constants:

  • $X$, a form of weight 2, with formula $X = ((\theta_{0000})^4+(\theta_{0001})^4+(\theta_{0010})^4+(\theta_{0011})^4)/4.$
  • $X(2\Omega)$, a form of weight 2.
  • $f_2(2\Omega)$, a form of weight 4, with formula $f_2 = (\theta_{0000})^4.$
  • $K(2\Omega)$, a form of of weight 6, with formula $$K = \theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111})^2/4096.$$
  • $f_{11}(2\Omega)$, a cusp form of weight 11, with formula $f_{11} = f_6\chi_5,$ where $$ \chi_5=\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011}\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111},\qquad f_6 = ((\theta_{0001})^{4}-(\theta_{0010})^{4}) ((\theta_{0001})^{4}-(\theta_{0011})^{4})((\theta_{0010})^{4}-(\theta_{0011})^{4}). $$

Note that we write $F(2\Omega)$ to mean "apply $F$ after doubling the input".

The generators $X, X(2\Omega),f_2(2\Omega), K(2\Omega)$ are algebraically independent. Let $B={\Bbb C}[X, X(2\Omega),f_2(2\Omega), K(2\Omega)]$. The module of modular forms with character of even weights only is then give by $$ \oplus_{k=0}^\infty M_{2k}(\Gamma_0(4),\psi_4) = f_{11}(2\Omega)f_{1}(2\Omega)B + Y(2\Omega) B + f_{11}(2\Omega)f_{3}(2\Omega)B, $$ where $f_1=\theta_{0000}^2$ and $f_3=(\theta_{0000}\theta_{0001}\theta_{0011})^2$.