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The energy spectrum of a hydrogen atom in a plasma has been of interest in physics until now. This problem is also quite important for astrophysics when considering the system in a magnetic field. This work suggests a method for numerically solving the Schrödinger equation of a plasma-embedded hydrogen atom in a uniform magnetic field using a more generalized exponential cosine screened Coulomb potential (MGECSC). The first special feature of the method is to convert the problem into an anharmonic oscillator by using the Kustaanheimo–Stiefel transformation. The second one is to exactly calculate the matrix elements concerning the harmonic oscillator basis set. These allow us to apply the Feranchuk–Komarov operator method to the Schrödinger equation for obtaining numerical solutions converging to any given precision. In this work, we obtain energies with a record precision of up to 30 decimal places for the ground and highly excited states with the principal quantum number up to n = 10. We test the FORTRAN program for a wide range of the magnetic field up to 10 a.u. (2.35 × 10 6 T), exceeding the threshold in the neutron stars. Also, the program works well with the range of the screening parameters describing the plasma environment in the previous theoretical and experimental studies. Apart from energies, the program also provides the corresponding wave functions. The results are meaningful not only for the development of methods but also for physics analysis and benchmarks for other approximate methods. [ABSTRACT FROM AUTHOR]