Rédei's triple symbols and modular forms

In 1939, L. Rédei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Rédei's triple symbol [a₁,a₂, p] describes the decomposition law of a prime number p in a certain dihedral extension over Q of degree 8 determined by a₁ and a₂. In this paper, we show that the triple symbol [-p₁, P₂, P₃] for certain prime numbers p ₁, p₂ and py can be expressed as a Fourier coefficient of a modular form of weight one. For this, we employ Hecke's theory on theta series associated to binary quadratic forms and realize an explicit version of the theorem by Weil-Langlands and Deligne-Serre for Rédei's dihedral extensions. A reciprocity law for the Rédei triple symbols yields certain reciprocal relations among Fourier coefficients.