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This paper is devoted to the interpolatory quadrature rules with an even number of multiple nodes, which have the maximal trigonometric degree of exactness. For the constructing of such quadrature rules we introduce and consider the so-called s- and $\sigma$-orthogonal trigonometric polynomials. We present a numerical method  for construction of mentioned quadrature rules and some numerical examples are also included.

$\sigma$-orthogonal trigonometric polynomials; Maximal trigonometric degree of exactness; Quadrature rules; Even number of multiple nodes