一. 写在前面

w1=(-1+sqrt(-3))/2; w2=(-1-sqrt(-3))/2;

exp(2*pi*j/5)=(-1+sqrt(5)+sqrt(-10-2*sqrt(5)))/4;

(资料图片)

exp(4*pi*j/5)=(-1-sqrt(5)+sqrt(-10+2*sqrt(5)))/4;

exp(6*pi*j/5)=(-1-sqrt(5)-sqrt(-10+2*sqrt(5)))/4; 

exp(8*pi*j/5)=(-1+sqrt(5)-sqrt(-10-2*sqrt(5)))/4;

j=sqrt(-1);

二. x^7=1的完整版复数解

2*cos(2*k*pi/7)，解x^3+x^2-2*x-1=0即得到。

2*sin(2*k*pi/7)，解x^3-sqrt(7)*x^2+sqrt(7)=0即得到。

exp(2*pi*j/7)=(-2+(28+84*sqrt(-3))^(1/3)+(28-84*sqrt(-3))^(1/3)+j*(2*sqrt(7)-w2*(52*sqrt(7)+12*sqrt(-21))^(1/3)-w1*(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

exp(4*pi*j/7)=(-2+w2*(28+84*sqrt(-3))^(1/3)+w1*(28-84*sqrt(-3))^(1/3)+j*(2*sqrt(7)-w1*(52*sqrt(7)+12*sqrt(-21))^(1/3)-w2*(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

exp(6*pi*j/7)=(-2+w1*(28+84*sqrt(-3))^(1/3)+w2*(28-84*sqrt(-3))^(1/3)-j*(2*sqrt(7)-(52*sqrt(7)+12*sqrt(-21))^(1/3)-(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

exp(8*pi*j/7)=(-2+w1*(28+84*sqrt(-3))^(1/3)+w2*(28-84*sqrt(-3))^(1/3)+j*(2*sqrt(7)-(52*sqrt(7)+12*sqrt(-21))^(1/3)-(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

exp(10*pi*j/7)=(-2+w2*(28+84*sqrt(-3))^(1/3)+w1*(28-84*sqrt(-3))^(1/3)-j*(2*sqrt(7)-w1*(52*sqrt(7)+12*sqrt(-21))^(1/3)-w2*(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

exp(12*pi*j/7)=(-2+(28+84*sqrt(-3))^(1/3)+(28-84*sqrt(-3))^(1/3)-j*(2*sqrt(7)-w2*(52*sqrt(7)+12*sqrt(-21))^(1/3)-w1*(52*sqrt(7)-12*sqrt(-21))^(1/3)))/12;

三. x^9=1的完整版复数解

exp(2*pi*j/9)=(-4+4*sqrt(-3))^(1/3)/2; exp(16*pi*j/9)=(-4-4*sqrt(-3))^(1/3)/2;

exp(8*pi*j/9)=w1*(-4+4*sqrt(-3))^(1/3)/2; exp(10*pi*j/9)=w2*(-4-4*sqrt(-3))^(1/3)/2;

exp(14*pi*j/9)=w2*(-4+4*sqrt(-3))^(1/3)/2; exp(4*pi*j/9)=w1*(-4-4*sqrt(-3))^(1/3)/2;

四. x^11=1的完整版复数解

exp(2*pi*j/11)=0.05*(-2-e3*(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-e2*(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-e3*(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-e2*(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2+j*(2*sqrt(11)+e3*(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+e2*(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

exp(4*pi*j/11)=0.05*(-2-e2*(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-e3*(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-e1*(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-e4*(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2-j*(2*sqrt(11)+e2*(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+e3*(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+e2*(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+e3*(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

exp(6*pi*j/11)=0.05*(-2-(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-e2*(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-e3*(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2+j*(2*sqrt(11)+(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+e1*(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+e4*(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

exp(8*pi*j/11)=0.05*(-2-e1*(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-e4*(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-e4*(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-e1*(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2+j*(2*sqrt(11)+e1*(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+e4*(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+e4*(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+e1*(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

exp(10*pi*j/11)=0.05*(-2-e4*(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-e1*(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2+j*(2*sqrt(11)+e4*(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+e1*(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+e3*(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+e2*(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

exp(12*pi*j/11)=0.05*(-2-e4*(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-e1*(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2-j*(2*sqrt(11)+e4*(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+e1*(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+e3*(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+e2*(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

exp(14*pi*j/11)=0.05*(-2-e1*(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-e4*(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-e4*(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-e1*(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2-j*(2*sqrt(11)+e1*(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+e4*(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+e4*(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+e1*(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

exp(16*pi*j/11)=0.05*(-2-(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-e2*(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-e3*(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2-j*(2*sqrt(11)+(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+e1*(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+e4*(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

exp(18*pi*j/11)=0.05*(-2-e2*(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-e3*(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-e1*(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-e4*(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2+j*(2*sqrt(11)+e2*(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+e3*(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+e2*(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+e3*(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

exp(20*pi*j/11)=0.05*(-2-e3*(7832+2200*sqrt(5)+440*sqrt(410-178*sqrt(5))*j)^0.2-e2*(7832+2200*sqrt(5)-440*sqrt(410-178*sqrt(5))*j)^0.2-e3*(7832-2200*sqrt(5)+440*sqrt(410+178*sqrt(5))*j)^0.2-e2*(7832-2200*sqrt(5)-440*sqrt(410+178*sqrt(5))*j)^0.2-j*(2*sqrt(11)+e3*(872*sqrt(11)+200*sqrt(55)+40*sqrt(96470-2398*sqrt(5))*j)^0.2+e2*(872*sqrt(11)+200*sqrt(55)-40*sqrt(96470-2398*sqrt(5))*j)^0.2+(872*sqrt(11)-200*sqrt(55)+40*sqrt(96470+2398*sqrt(5))*j)^0.2+(872*sqrt(11)-200*sqrt(55)-40*sqrt(96470+2398*sqrt(5))*j)^0.2));

五. x^13=1的完整版复数解

exp(2*pi*j/13)=(-1+sqrt(13)+(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+(104-20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26-6*sqrt(13))-w2*(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-w1*(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

exp(24*pi*j/13)=(-1+sqrt(13)+(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+(104-20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26-6*sqrt(13))-w2*(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-w1*(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

exp(6*pi*j/13)=(-1+sqrt(13)+w2*(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+w1*(104-20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26-6*sqrt(13))-w1*(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-w2*(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

exp(20*pi*j/13)=(-1+sqrt(13)+w2*(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+w1*(104-20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26-6*sqrt(13))-w1*(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-w2*(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

exp(18*pi*j/13)=(-1+sqrt(13)+w1*(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+w2*(104-20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26-6*sqrt(13))-(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

exp(8*pi*j/13)=(-1+sqrt(13)+w1*(104-20*sqrt(13)+12*sqrt(-39))^(1/3)+w2*(104-20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26-6*sqrt(13))-(20*sqrt(65+18*sqrt(13))+12*sqrt(-195-54*sqrt(13)))^(1/3)-(20*sqrt(65+18*sqrt(13))-12*sqrt(-195-54*sqrt(13)))^(1/3)))/12;

exp(10*pi*j/13)=(-1-sqrt(13)+w1*(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+w2*(104+20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26+6*sqrt(13))-w2*(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-w1*(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

exp(16*pi*j/13)=(-1-sqrt(13)+w1*(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+w2*(104+20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26+6*sqrt(13))-w2*(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-w1*(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

exp(4*pi*j/13)=(-1-sqrt(13)+(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+(104+20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26+6*sqrt(13))-w1*(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-w2*(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

exp(22*pi*j/13)=(-1-sqrt(13)+(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+(104+20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26+6*sqrt(13))-w1*(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-w2*(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

exp(12*pi*j/13)=(-1-sqrt(13)+w2*(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+w1*(104+20*sqrt(13)-12*sqrt(-39))^(1/3)+j*(sqrt(26+6*sqrt(13))-(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

exp(14*pi*j/13)=(-1-sqrt(13)+w2*(104+20*sqrt(13)+12*sqrt(-39))^(1/3)+w1*(104+20*sqrt(13)-12*sqrt(-39))^(1/3)-j*(sqrt(26+6*sqrt(13))-(20*sqrt(65-18*sqrt(13))+12*sqrt(-195+54*sqrt(13)))^(1/3)-(20*sqrt(65-18*sqrt(13))-12*sqrt(-195+54*sqrt(13)))^(1/3)))/12;

六. x^19=1的完整版复数解

具有三次根式双嵌套。

exp(2*pi*j/19)=(-2+w2*(532+228*sqrt(3)*j)^(1/3)+w1*(532-228*sqrt(3)*j)^(1/3)+(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))+162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)+(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))-162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)+j*(2*sqrt(19)+w2*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w1*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w2*(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w1*(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(36*pi*j/19)=(-2+w2*(532+228*sqrt(3)*j)^(1/3)+w1*(532-228*sqrt(3)*j)^(1/3)+(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))+162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)+(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))-162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)-j*(2*sqrt(19)+w2*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w1*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w2*(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w1*(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(14*pi*j/19)=(-2+w2*(532+228*sqrt(3)*j)^(1/3)+w1*(532-228*sqrt(3)*j)^(1/3)+w2*(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))+162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w1*(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))-162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)+j*(2*sqrt(19)+w2*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w1*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(24*pi*j/19)=(-2+w2*(532+228*sqrt(3)*j)^(1/3)+w1*(532-228*sqrt(3)*j)^(1/3)+w2*(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))+162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w1*(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))-162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)-j*(2*sqrt(19)+w2*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w1*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(22*pi*j/19)=(-2+w2*(532+228*sqrt(3)*j)^(1/3)+w1*(532-228*sqrt(3)*j)^(1/3)+w1*(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))+162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w2*(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))-162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)+j*(2*sqrt(19)+w2*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w1*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w1*(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w2*(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(16*pi*j/19)=(-2+w2*(532+228*sqrt(3)*j)^(1/3)+w1*(532-228*sqrt(3)*j)^(1/3)+w1*(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))+162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w2*(2736+18*((4215644+544236*sqrt(3)*j)^(1/3)+(4215644-544236*sqrt(3)*j)^(1/3))-162*j*((228*sqrt(3)+532*j)^(1/3)+(228*sqrt(3)-532*j)^(1/3)))^(1/3)-j*(2*sqrt(19)+w2*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w1*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w1*(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w2*(504*sqrt(19)+18*(w1*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w2*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w2*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w1*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(34*pi*j/19)=(-2+(532+228*sqrt(3)*j)^(1/3)+(532-228*sqrt(3)*j)^(1/3)+(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+j*(2*sqrt(19)+w1*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w2*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w2*(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w1*(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(4*pi*j/19)=(-2+(532+228*sqrt(3)*j)^(1/3)+(532-228*sqrt(3)*j)^(1/3)+(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)-j*(2*sqrt(19)+w1*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w2*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w2*(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w1*(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(10*pi*j/19)=(-2+(532+228*sqrt(3)*j)^(1/3)+(532-228*sqrt(3)*j)^(1/3)+w2*(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w1*(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+j*(2*sqrt(19)+w1*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w2*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(28*pi*j/19)=(-2+(532+228*sqrt(3)*j)^(1/3)+(532-228*sqrt(3)*j)^(1/3)+w2*(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w1*(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)-j*(2*sqrt(19)+w1*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w2*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(32*pi*j/19)=(-2+(532+228*sqrt(3)*j)^(1/3)+(532-228*sqrt(3)*j)^(1/3)+w1*(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w2*(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+j*(2*sqrt(19)+w1*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w2*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w1*(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w2*(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(6*pi*j/19)=(-2+(532+228*sqrt(3)*j)^(1/3)+(532-228*sqrt(3)*j)^(1/3)+w1*(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w2*(2736+18*(w1*(4215644+544236*sqrt(3)*j)^(1/3)+w2*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w2*(228*sqrt(3)+532*j)^(1/3)+w1*(228*sqrt(3)-532*j)^(1/3)))^(1/3)-j*(2*sqrt(19)+w1*(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+w2*(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w1*(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w2*(504*sqrt(19)+18*((1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-w1*(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-w2*(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(8*pi*j/19)=(-2+w1*(532+228*sqrt(3)*j)^(1/3)+w2*(532-228*sqrt(3)*j)^(1/3)+(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+j*(2*sqrt(19)+(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(30*pi*j/19)=(-2+w1*(532+228*sqrt(3)*j)^(1/3)+w2*(532-228*sqrt(3)*j)^(1/3)+(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)-j*(2*sqrt(19)+(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(12*pi*j/19)=(-2+w1*(532+228*sqrt(3)*j)^(1/3)+w2*(532-228*sqrt(3)*j)^(1/3)+w1*(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w2*(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+j*(2*sqrt(19)+(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w2*(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w1*(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(26*pi*j/19)=(-2+w1*(532+228*sqrt(3)*j)^(1/3)+w2*(532-228*sqrt(3)*j)^(1/3)+w1*(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w2*(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)-j*(2*sqrt(19)+(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w2*(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w1*(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(18*pi*j/19)=(-2+w1*(532+228*sqrt(3)*j)^(1/3)+w2*(532-228*sqrt(3)*j)^(1/3)+w2*(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w1*(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+j*(2*sqrt(19)+(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w1*(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w2*(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

exp(20*pi*j/19)=(-2+w1*(532+228*sqrt(3)*j)^(1/3)+w2*(532-228*sqrt(3)*j)^(1/3)+w2*(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))+162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)+w1*(2736+18*(w2*(4215644+544236*sqrt(3)*j)^(1/3)+w1*(4215644-544236*sqrt(3)*j)^(1/3))-162*j*(w1*(228*sqrt(3)+532*j)^(1/3)+w2*(228*sqrt(3)-532*j)^(1/3)))^(1/3)-j*(2*sqrt(19)+(44*sqrt(19)+84*sqrt(57)*j)^(1/3)+(44*sqrt(19)-84*sqrt(57)*j)^(1/3)+w1*(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))+54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3)+w2*(504*sqrt(19)+18*(w2*(1225684*sqrt(19)+57372*sqrt(57)*j)^(1/3)+w1*(1225684*sqrt(19)-57372*sqrt(57)*j)^(1/3))-54*j*(4*sqrt(57)-(132*sqrt(57)+756*sqrt(19)*j)^(1/3)-(132*sqrt(57)-756*sqrt(19)*j)^(1/3)))^(1/3))/36;

七. 写在最后

不难发现，以上结果具有很优美的对称性。随着N的增大，表达式越来越复杂，而且节点也更丰富。经过Magma数值验证，只具有数值上的细微误差。

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