Context. Weak gravitational lensing is a key cosmological probe for current and future large-scale surveys. While power spectra are commonly used for analyses, they fail to capture non-Gaussian information from nonlinear structure formation, necessitating higher-order statistics and methods for efficient map generation. Aims. To develop an emulator that generates accurate convergence (κ) maps directly from an input power spectrum and wavelet ℓ1-norm without relying on computationally intensive simulations. Methods. We use either numerical or theoretical predictions to construct κ maps by iteratively adjusting wavelet coefficients to match target marginal distributions and their inter-scale dependencies, incorporating higher-order statistical information. Results. The resulting κ maps accurately reproduce the input power spectrum and exhibit higher-order statistical properties consistent with the input predictions, providing an efficient tool for weak lensing analyses.