In this paper,we consider the dimension and the basis of the space of the Siegel Eisenstein series of degree 2,weight 2,on the congruence subgroup of squarefree level N.We determine the arithmetic relation between the Siegel Eisenstein series and theta series associated with the order of the positive definite quaternion algebra.Firstly,we construct a full set of the basis of the space of squarefree level Siegel Eisenstein series and obtain their exact formulas for Fourier coefficients by using the modified Hurwitz class number.Secondly,for the theta series defined by Yoshida on the Eichler order of a positive definite quaternion algebra over the rational number field Q,we compute its Fourier coefficients and the Siegel-Weil type formula explicitly.
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