Summary: We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose-Einstein condensates. This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum \(n(\omega)\) at the zero frequency \(\omega\). The self-similar solution is of the second kind, and it satisfies boundary conditions corresponding to a nonzero constant spectrum (with all its derivative being zero) at \(\omega=0\) and a power-law asymptotic \(n(\omega)\to\omega^{-x}\) at \(\omega\to\infty x\in\mathbb{R}^+\). Finding it amounts to solving a nonlinear eigenvalue problem, i.e. finding the value \(x^*\) of the exponent \(x\) for which these two boundary conditions can be satisfied simultaneously. To solve this problem we develop a new high-precision algorithm based on Chebyshev approximations and double exponential formulas for evaluating the collision integral, as well as the iterative techniques for solving the integro-differential equation for the self-similar shape function. This procedures allow to achieve a solution with accuracy \(\approx 4.7\%\) which is realized for \(x^*\approx 1.22\).