The authors prove a general form of the sum formula $mathrm{SL}_2$ over a totally real number field. This formula relates sums of Kloosterman sums to products of Fourier coefficients of automorphic representations. The authors give two the spectral sum formula (in sum formula) and the Kloosterman sum formula. They have the independent test function in the spectral term, in the sum of Kloosterman sums, respectively.
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