In connection with Eisenstein series for the principal congruence subgroup \(\Gamma (n)\) , Hecke introduced certain numbers, of which he said that they are rational and cumbersome to calculate. We show, however, that these numbers are (essentially) generators of the nth cyclotomic field or its maximal real subfield. They arise from the well investigated cotangent numbers by matrix inversion, which is why we call them inverse cotangent numbers. We describe them as linear combinations of roots of unity with rational coefficients in a fairly closed form, provided that n is square-free. We also exhibit a formula for these numbers in terms of generalized Bernoulli numbers and Gauss sums.
