Θ[a_, b_, ord_] := Θ[a + 2, b, ord] /; a < 0;
Θ[a_, b_, ord_] := Θ[a - 2, b, ord] /; a > 2;
Θ[a_, b_, ord_] := Exp[-I Pi a]*Θ[a, b + 2, ord] /; b < 0;
Θ[a_, b_, ord_] := Exp[I Pi a]*Θ[a, b - 2, ord] /; b >= 2;
Θ[a_, b_, ord_] := (qi qr)^(a^2/4) Exp[I Pi a b/2] + Sum[(qi qr)^(n + a/2)^2 Exp[I Pi (n + a/2) b] + (qr*qi)^(n - a/2)^2 Exp[I Pi (-n + a/2) b] , {n, 1,ord}]+O[qr]^((ord + 1 - Abs[a]/2)^2);
Θc[a_, b_, ord_] := Θc[a + 2, b, ord] /; a < 0;
Θc[a_, b_, ord_] := Θc[a - 2, b, ord] /; a > 2;
Θc[a_, b_, ord_] := Exp[I Pi a]*Θc[a, b + 2, ord] /; b < 0;
Θc[a_, b_, ord_] := Exp[-I Pi a]*Θc[a, b - 2, ord] /; b >= 2;
Θc[a_, b_, ord_] := (qr/qi)^(a^2/4) Exp[I Pi a b/2] + Sum[(qr/qi)^(n+a/2)^2Exp[I Pi (n + a/2) b] + (qr/qi)^(n-a/2)^2 Exp[I Pi (-n+a/2) b] , {n, 1, ord}]+O[qr]^((ord + 1 - Abs[a]/2)^2);

ord=5;

Z1 = 2*(Θ[1,0,ord]^4*(Θc[0,0,ord]^8 Θc[0,1,ord]^8+Θc[0,1,ord]^8 Θc[0,0,ord]^8));
Z1//FullSimplify

64 qi qr+(256 (-8+qi^4) qr^3)/qi+(128 (240-64 qi^4+3 qi^8) qr^5)/qi^3