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The elliptic modular j function enjoys many beautiful properties. Its Fouriercoefficients are related to the Monster group, and its CM values generateabelian extensions over imaginary quadratic fields. Kaneko gave an arithmeticformula for the Fourier coefficients expressed in terms of the traces of the CMvalues. In this article, we are concerned with analogues of Kaneko's result forthe McKay-Thompson series of square-free level.

The Fourier coefficients of the McKay-Thompson series and the traces of CM values