(****************************** GAMMAs120, Version 1.13m (5 April, 2004) ********************************

by  R. Vidunas, Kyushu Univercity; Supported by the 21st Century COE Program
This is a small Mathematica package for simplifying products and quotients of Gamma values.
The package has to be read with this commando:
<< gammas120.m

Some combinations of Gamma values occur naturally as algebraic numbers or values of
hypergeometric functions. They can be simplified using the classical recurrence equation,
relfection and multiplication formulas. This package can simplify products and quotients
of Gamma values Gamma(x) at the rational points x suc that the denominator of x divides 120.
The expressions are products or quotients of the 16 "basic' gamma values at
1/3, 1/4, 1/5, 2/5, 1/8, 1/15, 1/20, 1/24, 1/40, 3/40, 7/40, 1/60, 7/60, 1/120, 7/120, 11/120
and rational powers of Pi and several algebraic numbers.

A related paper is:
 R. Vidunas, 'Expressions for values of the gamma function', http://arxiv.org/math.CA/0403510
 
************************************************************************************************)

BeginPackage["Gamma120`"]

GammaSimplify::usage = 
"GammaSimplify[GT] simplifies the gamma term GT by evaluating the gamma values
 at the points k/120 (with integer k) and trigonometric values at k*Pi/120
 in terms of the gamma values in the list GammaValuesBasis, and rational powers
 of Pi and algebraic numbers in the list GammaNumbers. Some algebraic numbers
 are represented compactly; to expand them fully, see GammaPsis."

GammaPsis::usage = 
"The list of substitutions for rewriting some algebraic constants (returned by GammaSimplify)
 explicitly. It has to be used, for example, before numeric evaluation.
 Warning: due to possible Mathematica iteration limitations, numeric evaluation may be incomplete.
 In these cases, use ReleaseHold[], or change $IterationLimit." 

Gamma2EllipticK::usage =
"The list for substititions for expressing the Gamma values at 1/3, 1/4, 1/8, 1/15, 1/20, 1/24 in
 terms of the elliptic K function. Can be used after GammaSimplify[] to express gamma terms similarly."

Gammas::usage =
"The list of substitutions for gamma values k/120 with k=1,2,...,119, in terms
 of the gamma values in the list GammaValuesBasis, and rational powers of the
 numbers in the list GammaNumbers." 
               
GammaValuesBasis::usage = 
"The list of 16 gamma values which can be used to express all gamma values at
 the points k/120, with integer k. See GammaSimplify[]."
                
GammaNumbers::usage =
"The list containing Pi and algebraic numbers which are used in GammaSimplify[]. Some algebraic
 constants are represented compactly, to expand them fully, see the substitution list GammaPsis.
 For some relations between the algebraic values, see GammaAlgebraicIdentities."

GammaAlgebraicIdentities::usage =
"The list of useful identities between the algebraic numbers in the list GammaNumbers.
 This list is not quite ready to be used automatically."

G\[Phi]::usage = "Represents the constant 5+Sqrt[5]."
g\[Phi]::usage = "Represents the constant 5-Sqrt[5]."
G\[Psi]::usage = "Represents the constant Sqrt[5+2 Sqrt[5]]"
g\[Psi]::usage = "Represents the constant Sqrt[5-2 Sqrt[5]]"

Begin["Private`"]

Print["GAMMAs120, Version 1.13m (5 April, 2004),"]
Print["by  R. Vidunas, Kyushu Univercity; Supported by the 21st Century COE Programme."];
Print["This is a small Mathematica package for simplifying products and quotients of Gamma values."];
Print["Please see the starting comments in gammas120.m, or type ?GammaSimplify, or"];
Print["or ?GammaPsis, ?Gamma2EllipticK, ?Gammas, ?GammasValuesBasis, ?GammaNumbers,"];
Print["?G\[Phi], ?g\[Phi], ?G\[Psi], ?g\[Psi], ?GammaAlgebraicIdentities"];

GammaSimplify[T_]:= 
  If[ Head[T]===Product, Map[GammaTrigoAdjust,T], GammaTrigoAdjust[T] ] /. Gammas; 

GammaTrigoAdjust[T_]:= Module[ {k, Q=First[T], QQ},
  If[ Head[T]===Power,  GammaTrigoAdjust[Q]^T[[2]],
    If[ Head[T]===Gamma,    
      k=Floor[Q];  QQ=Q-k; Pochhammer[QQ,k]Gamma[QQ],
      Q=Q/Pi;
      If[ Head[Q]===Rational && Mod[120,Denominator[Q]]==0,
        Switch[ Head[T], Csc, Gamma[Q]Gamma[1-Q]/Pi,
          Sin, Pi/Gamma[Q]/Gamma[1-Q],
          Sec, Gamma[1/2+Q]Gamma[1/2-Q]/Pi,
          Cos, Pi/Gamma[1/2+Q]/Gamma[1/2-Q], _, T ],
        T ]
] ] ];

GammaValuesBasis =  { Gamma[1/3], Gamma[1/4], Gamma[1/5], Gamma[2/5], Gamma[1/8], Gamma[1/15], Gamma[1/20], Gamma[1/24],
  Gamma[1/40], Gamma[3/40], Gamma[7/40], Gamma[1/60], Gamma[7/60], Gamma[1/120], Gamma[7/120], Gamma[11/120] };

GammaNumbers = { Pi, 2, 3, 5, Sqrt[2]+1, Sqrt[3]+1, Sqrt[3]+Sqrt[2], Sqrt[5]+Sqrt[3], Sqrt[6]+Sqrt[5], Sqrt[10]+3,
G\[Phi], Sqrt[10]+Sqrt[G\[Phi]], Sqrt[10]+Sqrt[g\[Phi]], G\[Psi]+Sqrt[15], g\[Psi]+Sqrt[15],
Sqrt[G\[Phi]]+Sqrt[5], Sqrt[g\[Phi]]+Sqrt[5], Sqrt[G\[Phi]]+Sqrt[3], Sqrt[g\[Phi]]+Sqrt[3] };

Gammas = { Gamma[2/3]->2*Pi/Sqrt[3]/Gamma[1/3],
Gamma[3/4]->Pi*Sqrt[2]/Gamma[1/4],
Gamma[3/5]->Pi*2^(3/2)/Sqrt[G\[Phi]]/Gamma[2/5],
Gamma[4/5]->Pi*Sqrt[2]*Sqrt[G\[Phi]]/Sqrt[5]/Gamma[1/5],
Gamma[1/6]->Sqrt[3]/2^(1/3)/Sqrt[Pi]*Gamma[1/3]^2,
Gamma[5/6]->Pi^(3/2)*2^(4/3)/Sqrt[3]/Gamma[1/3]^2,
Gamma[3/8]->Sqrt[Pi]/Sqrt[Sqrt[2]+1]/Gamma[1/4]*Gamma[1/8],
Gamma[5/8]->Sqrt[Pi]*2^(3/4)*Gamma[1/4]/Gamma[1/8],
Gamma[7/8]->Pi*2^(3/4)*Sqrt[Sqrt[2]+1]/Gamma[1/8],
Gamma[1/10]->Sqrt[G\[Phi]]/Sqrt[Pi]/2^(7/10)*Gamma[1/5]*Gamma[2/5],
Gamma[3/10]->Sqrt[Pi]*2^(7/5)*Sqrt[5]/G\[Phi]*Gamma[1/5]/Gamma[2/5],
Gamma[7/10]->Sqrt[Pi]*2^(3/5)*Gamma[2/5]/Gamma[1/5],
Gamma[9/10]->Pi^(3/2)*2^(7/10)*Sqrt[G\[Phi]]/Sqrt[5]/Gamma[1/5]/Gamma[2/5],
Gamma[1/12]->3^(3/8)/Sqrt[Pi]/2^(1/4)*Sqrt[Sqrt[3]+1]*Gamma[1/3]*Gamma[1/4],
Gamma[5/12]->Sqrt[Pi]*2^(3/4)/3^(1/8)/Sqrt[Sqrt[3]+1]/Gamma[1/3]*Gamma[1/4],
Gamma[7/12]->Sqrt[Pi]*2^(3/4)*3^(1/8)/Sqrt[Sqrt[3]+1]*Gamma[1/3]/Gamma[1/4],
Gamma[11/12]->Pi^(3/2)*2^(3/4)/3^(3/8)*Sqrt[Sqrt[3]+1]/Gamma[1/3]/Gamma[1/4],
Gamma[2/15]->Sqrt[2]/3^(7/20)*5^(1/6)/Sqrt[g\[Psi]+Sqrt[15]]/Gamma[1/3]*Gamma[2/5]*Gamma[1/15],
Gamma[4/15]->Sqrt[G\[Phi]]*Sqrt[2]/3^(3/10)/Sqrt[G\[Psi]+Sqrt[15]]/Sqrt[g\[Psi]+Sqrt[15]]/Gamma[1/5]*Gamma[2/5]*Gamma[1/15],
Gamma[7/15]->3^(9/20)*5^(1/3)/Sqrt[G\[Phi]]*Sqrt[g\[Psi]+Sqrt[15]]*Gamma[1/3]*Gamma[1/5]/Gamma[1/15],
Gamma[8/15]->Pi*4/3^(9/20)*5^(1/6)/Sqrt[G\[Psi]+Sqrt[15]]/Sqrt[G\[Phi]]/Gamma[1/3]/Gamma[1/5]*Gamma[1/15],
Gamma[11/15]->2*Pi*3^(3/10)*Gamma[1/5]/Gamma[2/5]/Gamma[1/15],
Gamma[13/15]->Pi*Sqrt[2]*3^(7/20)*Sqrt[G\[Psi]+Sqrt[15]]/5^(1/6)*Gamma[1/3]/Gamma[2/5]/Gamma[1/15],
Gamma[14/15]->Pi/Sqrt[2]/Sqrt[5]*Sqrt[G\[Phi]]*Sqrt[G\[Psi]+Sqrt[15]]*Sqrt[g\[Psi]+Sqrt[15]]/Gamma[1/15],
Gamma[3/20]->Sqrt[Pi]*2^(19/20)*5^(1/8)/Sqrt[G\[Phi]]/Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]/Gamma[2/5]*Gamma[1/20],
Gamma[7/20]->Sqrt[Pi]*2^(17/20)*5^(1/8)/Sqrt[G\[Phi]]/Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]/Gamma[1/5]*Gamma[1/20],
Gamma[9/20]->Pi*2^(4/5)/Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]/Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]/Gamma[1/5]/Gamma[2/5]*Gamma[1/20],
Gamma[11/20]->2^(1/5)*Sqrt[G\[Phi]]*Gamma[1/5]*Gamma[2/5]/Gamma[1/20],
Gamma[13/20]->Sqrt[Pi]*2^(23/20)*5^(3/8)/Sqrt[G\[Phi]]*Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*Gamma[1/5]/Gamma[1/20],
Gamma[17/20]->Sqrt[Pi]*2^(1/20)*Sqrt[G\[Phi]]*Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]/5^(1/8)*Gamma[2/5]/Gamma[1/20],
Gamma[19/20]->Pi/Sqrt[5]*Sqrt[G\[Phi]]*Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]*Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]/Gamma[1/20],
Gamma[5/24]->Sqrt[Pi]*2^(1/3)/Sqrt[3]/Sqrt[Sqrt[2]+1]/Sqrt[Sqrt[3]+1]/Gamma[1/3]*Gamma[1/24],
Gamma[7/24]->Sqrt[Pi]*2^(1/4)/3^(3/8)/Sqrt[Sqrt[3]+1]/Sqrt[Sqrt[3]+Sqrt[2]]/Gamma[1/4]*Gamma[1/24],
Gamma[11/24]->Pi*2^(1/12)/3^(3/8)/Sqrt[Sqrt[2]+1]/Sqrt[Sqrt[3]+Sqrt[2]]/Gamma[1/3]/Gamma[1/4]*Gamma[1/24],
Gamma[13/24]->2^(2/3)*3^(3/8)*Sqrt[Sqrt[3]+1]*Gamma[1/3]*Gamma[1/4]/Gamma[1/24],
Gamma[17/24]->2*Sqrt[Pi]*3^(3/8)*Sqrt[Sqrt[2]+1]*Gamma[1/4]/Gamma[1/24],
Gamma[19/24]->Sqrt[Pi]*2^(11/12)*Sqrt[3]*Sqrt[Sqrt[3]+Sqrt[2]]*Gamma[1/3]/Gamma[1/24],
Gamma[23/24]->Pi*2^(3/4)*Sqrt[Sqrt[2]+1]*Sqrt[Sqrt[3]+1]*Sqrt[Sqrt[3]+Sqrt[2]]/Gamma[1/24],
Gamma[1/30]->3^(9/20)/Sqrt[Pi]/2^(16/15)/5^(1/6)*Sqrt[G\[Phi]]*Sqrt[G\[Psi]+Sqrt[15]]*Gamma[1/3]*Gamma[1/5],
Gamma[7/30]->3^(3/20)/Sqrt[Pi]/2^(7/15)*5^(1/3)/Sqrt[G\[Phi]]*Sqrt[g\[Psi]+Sqrt[15]]*Gamma[1/3]*Gamma[2/5],
Gamma[11/30]->Sqrt[Pi]*2^(23/30)/3^(1/20)*5^(1/6)/Sqrt[G\[Psi]+Sqrt[15]]/Gamma[1/3]*Gamma[1/5],
Gamma[13/30]->Sqrt[Pi]*2^(17/15)*3^(7/20)*5^(1/3)/Sqrt[G\[Phi]]/Sqrt[g\[Psi]+Sqrt[15]]*Gamma[1/3]/Gamma[2/5],
Gamma[17/30]->Sqrt[Pi]*2^(41/30)/3^(7/20)*5^(1/6)/Sqrt[g\[Psi]+Sqrt[15]]/Gamma[1/3]*Gamma[2/5],
Gamma[19/30]->Sqrt[Pi]*2^(11/15)*3^(1/20)/5^(1/6)*Sqrt[G\[Phi]]/Sqrt[G\[Psi]+Sqrt[15]]*Gamma[1/3]/Gamma[1/5],
Gamma[23/30]->Pi^(3/2)*2^(59/30)/3^(3/20)*5^(1/6)/G\[Phi]*Sqrt[g\[Psi]+Sqrt[15]]/Gamma[1/3]/Gamma[2/5],
Gamma[29/30]->Pi^(3/2)/2^(13/30)/3^(9/20)/5^(5/6)*G\[Phi]*Sqrt[G\[Psi]+Sqrt[15]]/Gamma[1/3]/Gamma[1/5],
Gamma[9/40]->Sqrt[Pi]*2^(11/20)*5^(1/4)*Sqrt[Sqrt[10]+3]/Sqrt[Sqrt[2]+1]/Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*
  Gamma[1/8]^2*Gamma[1/20]/Gamma[1/40]*Gamma[7/40]/Gamma[2/5]/Gamma[3/40]/Gamma[1/4],
Gamma[11/40]->2^(1/5)*5^(1/8)*G\[Phi]^(1/4)/(Sqrt[2]+1)^(3/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[10]+3)^(1/4)*
  Sqrt[Sqrt[G\[Phi]]+Sqrt[5]]/(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*Gamma[1/8]^2*Gamma[1/5]*Gamma[7/40]/Gamma[1/4]/Gamma[3/40]/Gamma[1/40],
Gamma[13/40]->5^(1/4)/(Sqrt[2]+1)^(1/4)/Sqrt[Sqrt[g\[Phi]]+Sqrt[5]]*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*
  (Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[10]+3)^(1/4)*Gamma[1/5]*Gamma[7/40]/Gamma[1/20],
Gamma[17/40]->Sqrt[G\[Phi]]/2^(4/5)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[2]+1)^(1/4)/(Sqrt[10]+3)^(1/4)*
  (Sqrt[g\[Phi]]+Sqrt[10])^(1/4)/Sqrt[Sqrt[g\[Phi]]+Sqrt[5]]*Gamma[2/5]*Gamma[3/40]/Gamma[1/20],
Gamma[19/40]->Sqrt[Pi]/2^(9/20)/5^(1/8)*G\[Phi]^(1/4)/(Sqrt[2]+1)^(1/4)/(Sqrt[10]+3)^(1/4)/Sqrt[Sqrt[G\[Phi]]+Sqrt[5]]*
  (Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*Gamma[1/40]/Gamma[1/20],
Gamma[21/40]->Sqrt[Pi]*2^(19/20)*Gamma[1/20]/Gamma[1/40],
Gamma[23/40]->Pi*2^(9/5)*5^(1/8)/Sqrt[G\[Phi]]/Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*Gamma[1/20]/Gamma[2/5]/Gamma[3/40],
Gamma[27/40]->Pi*2^(3/2)*5^(1/8)/Sqrt[G\[Phi]]/Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]*Gamma[1/20]/Gamma[1/5]/Gamma[7/40],
Gamma[29/40]->Pi*2^(4/5)/5^(1/4)*Sqrt[Sqrt[2]+1]/Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]/Sqrt[Sqrt[10]+3]*
  Gamma[1/4]*Gamma[1/40]*Gamma[3/40]/Gamma[1/8]^2/Gamma[1/5]/Gamma[7/40],
Gamma[31/40]->Sqrt[Pi]*2^(9/20)/5^(1/8)*(Sqrt[2]+1)^(3/4)*G\[Phi]^(1/4)*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/
  (Sqrt[10]+3)^(1/4)/Sqrt[Sqrt[G\[Phi]]+Sqrt[5]]*Gamma[1/4]*Gamma[2/5]*Gamma[3/40]*Gamma[1/40]/Gamma[1/8]^2/Gamma[7/40]/Gamma[1/20],
Gamma[33/40]->Pi*2^(3/2)*5^(1/8)*(Sqrt[2]+1)^(1/4)/Sqrt[G\[Phi]]/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*
  (Sqrt[g\[Phi]]+Sqrt[10])^(1/4)/(Sqrt[10]+3)^(1/4)*Sqrt[Sqrt[g\[Phi]]+Sqrt[5]]/Gamma[7/40],
Gamma[37/40]->Pi*2/5^(1/8)/(Sqrt[2]+1)^(1/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/
  (Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[10]+3)^(1/4)*Sqrt[Sqrt[g\[Phi]]+Sqrt[5]]/Gamma[3/40],
Gamma[39/40]->Pi*Sqrt[2]/5^(3/8)*(Sqrt[2]+1)^(1/4)*G\[Phi]^(1/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*
  (Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[10]+3)^(1/4)*Sqrt[Sqrt[G\[Phi]]+Sqrt[5]]/Gamma[1/40],
Gamma[11/60]->Sqrt[Pi]*2^(5/4)/Sqrt[3]*5^(7/24)/Sqrt[G\[Phi]]/Sqrt[G\[Psi]+Sqrt[15]]/Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]*Gamma[1/60]/Gamma[1/3],
Gamma[13/60]->Sqrt[Pi]*2^(7/10)/3^(3/20)*5^(1/8)*Sqrt[Sqrt[3]+1]/Sqrt[Sqrt[5]+Sqrt[3]]/Sqrt[g\[Psi]+Sqrt[15]]*Gamma[7/60]/Gamma[2/5],
Gamma[17/60]->Sqrt[Pi]*2^(3/4)/Sqrt[3]*5^(1/24)*Sqrt[G\[Phi]]/Sqrt[g\[Psi]+Sqrt[15]]/Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*Gamma[7/60]/Gamma[1/3],
Gamma[19/60]->Sqrt[Pi]*2^(11/10)/3^(9/20)/5^(1/8)/Sqrt[Sqrt[3]+1]/Sqrt[Sqrt[5]+Sqrt[3]]/Sqrt[G\[Psi]+Sqrt[15]]*Gamma[1/60]/Gamma[1/5],
Gamma[23/60]->Pi*2^(19/20)/3^(3/20)/5^(1/12)*Sqrt[Sqrt[3]+1]/Sqrt[Sqrt[5]+Sqrt[3]]/Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*Gamma[7/60]/Gamma[2/5]/Gamma[1/3],
Gamma[29/60]->Pi*2^(17/20)/3^(9/20)/5^(1/12)/Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]/Sqrt[Sqrt[3]+1]/Sqrt[Sqrt[5]+Sqrt[3]]*Gamma[1/60]/Gamma[1/5]/Gamma[1/3],
Gamma[31/60]->3^(9/20)/2^(1/10)/5^(1/6)*Sqrt[G\[Phi]]*Sqrt[G\[Psi]+Sqrt[15]]*Gamma[1/5]*Gamma[1/3]/Gamma[1/60],
Gamma[37/60]->3^(3/20)*2^(3/10)*5^(1/3)/Sqrt[G\[Phi]]*Sqrt[g\[Psi]+Sqrt[15]]*Gamma[1/3]*Gamma[2/5]/Gamma[7/60],
Gamma[41/60]->Sqrt[Pi]*2^(3/20)*3^(9/20)/5^(1/8)*Sqrt[G\[Phi]]*Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]*Gamma[1/5]/Gamma[1/60],
Gamma[43/60]->Sqrt[Pi]*2*Sqrt[3]*5^(5/24)/Sqrt[G\[Phi]]/Sqrt[Sqrt[3]+1]*Sqrt[Sqrt[5]+Sqrt[3]]*Gamma[1/3]/Gamma[7/60],
Gamma[47/60]->Sqrt[Pi]*2^(21/20)*3^(3/20)*5^(1/8)/Sqrt[G\[Phi]]*Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*Gamma[2/5]/Gamma[7/60],
Gamma[49/60]->Sqrt[Pi]*Sqrt[3]/5^(1/24)*Sqrt[G\[Phi]]*Sqrt[Sqrt[3]+1]*Sqrt[Sqrt[5]+Sqrt[3]]*Gamma[1/3]/Gamma[1/60],
Gamma[53/60]->Pi*2^(5/4)*5^(1/4)/G\[Phi]/Sqrt[Sqrt[3]+1]*Sqrt[Sqrt[5]+Sqrt[3]]*Sqrt[g\[Psi]+Sqrt[15]]*Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]/Gamma[7/60],
Gamma[59/60]->Pi/2^(5/4)/5^(3/4)*G\[Phi]*Sqrt[Sqrt[3]+1]*Sqrt[Sqrt[5]+Sqrt[3]]*Sqrt[G\[Psi]+Sqrt[15]]*Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]/Gamma[1/60],
Gamma[13/120]->3^(7/40)/2^(53/120)/5^(1/8)*Sqrt[Sqrt[10]+Sqrt[G\[Phi]]]*Sqrt[Sqrt[10]+3]*G\[Phi]*(Sqrt[6]+Sqrt[5])^(1/4)*
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[3]+1)^(1/4)*(Sqrt[3]+Sqrt[g\[Phi]])^(1/4)/(Sqrt[2]+1)^(3/4)/(g\[Psi]+Sqrt[15])^(1/4)/
  (Sqrt[10]+Sqrt[g\[Phi]])^(1/4)/(Sqrt[5]+Sqrt[g\[Phi]])^(1/4)/(Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*
  Gamma[7/60]*Gamma[1/8]^2*Gamma[1/5]*Gamma[7/40]^2/Gamma[7/120]/Gamma[1/4]/Gamma[2/5]/Gamma[3/40]/Gamma[1/40],
Gamma[17/120]->(Sqrt[2]+1)/5^(1/8)/3^(1/4)/2^(1/30)*(Sqrt[10]+Sqrt[g\[Phi]])^(1/4)*(G\[Psi]+Sqrt[15])^(1/4)/G\[Phi]^(1/4)/
  (Sqrt[10]+Sqrt[G\[Phi]])^(1/4)/Sqrt[Sqrt[10]+3]/Sqrt[Sqrt[3]+Sqrt[g\[Phi]]]/Sqrt[Sqrt[5]+Sqrt[g\[Phi]]]/(g\[Psi]+Sqrt[15])^(1/4)*
  Gamma[2/5]*Gamma[3/40]^2*Gamma[11/120]*Gamma[7/120]*Gamma[1/120]*Gamma[1/4]/Gamma[1/8]^2/Gamma[1/60]/Gamma[1/5]/Gamma[7/40]^2,
Gamma[19/120]->3^(1/40)*2^(19/120)*G\[Phi]^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)*(Sqrt[10]+Sqrt[G\[Phi]])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*
  (Sqrt[3]+Sqrt[g\[Phi]])^(1/4)/(Sqrt[3]+1)^(1/4)/(Sqrt[2]+1)^(1/4)/(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(G\[Psi]+Sqrt[15])^(1/4)/
  (Sqrt[5]+Sqrt[g\[Phi]])^(1/4)/(Sqrt[5]+Sqrt[3])^(1/4)*Gamma[1/40]*Gamma[1/60]*Gamma[7/40]/Gamma[1/20]/Gamma[1/120],
Gamma[23/120]->Sqrt[Pi]/2^(37/120)/3^(13/40)/Sqrt[5]*(Sqrt[2]+1)^(3/4)*(G\[Psi]+Sqrt[15])^(1/4)*(Sqrt[3]+1)^(1/4)/Sqrt[Sqrt[10]+3]/
  (Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4)/(Sqrt[10]+Sqrt[G\[Phi]])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[3]+Sqrt[g\[Phi]])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[5]+Sqrt[g\[Phi]])^(1/4)*Gamma[7/120]*Gamma[1/4]*Gamma[1/40]*Gamma[3/40]*Gamma[11/120]*Gamma[1/120]/
  Gamma[1/8]^2/Gamma[1/5]/Gamma[1/60]/Gamma[7/40]^2,
Gamma[29/120]->(Sqrt[2]+1)^(3/4)/2^(9/8)/3^(9/40)/5^(3/8)*Sqrt[G\[Phi]]*Sqrt[Sqrt[10]+Sqrt[g\[Phi]]]*(Sqrt[6]+Sqrt[5])^(1/4)*
  (G\[Psi]+Sqrt[15])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/Sqrt[Sqrt[10]+3]/(Sqrt[3]+1)^(1/4)/(Sqrt[5]+Sqrt[3])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*
  Gamma[1/4]*Gamma[1/40]*Gamma[3/40]^2*Gamma[2/5]*Gamma[11/120]/Gamma[1/8]^2/Gamma[1/5]/Gamma[7/40]/Gamma[1/20],
Gamma[31/120]->3^(9/40)/Sqrt[Pi]/2^(97/60)/5^(7/24)*(Sqrt[2]+1)*G\[Phi]^(3/4)*Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*
  Sqrt[G\[Psi]+Sqrt[15]]/Sqrt[Sqrt[10]+3]/Sqrt[Sqrt[G\[Phi]]+Sqrt[5]]/Sqrt[Sqrt[g\[Phi]]+Sqrt[5]]*Gamma[1/3]*
  Gamma[1/4]*Gamma[2/5]*Gamma[3/40]^2*Gamma[1/40]*Gamma[11/120]/Gamma[1/8]^2/Gamma[7/40]/Gamma[1/20]/Gamma[1/60],
Gamma[37/120]->5^(1/12)/3^(7/40)*Sqrt[Sqrt[2]+1]/G\[Phi]*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*(g\[Psi]+Sqrt[15])^(1/4)*(G\[Psi]+Sqrt[15])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/Sqrt[Sqrt[10]+3]/Sqrt[Sqrt[G\[Phi]]+Sqrt[5]]/Sqrt[Sqrt[g\[Phi]]+Sqrt[3]]*Gamma[1/3]*Gamma[1/4]*
  Gamma[7/120]*Gamma[2/5]*Gamma[3/40]*Gamma[1/40]*Gamma[1/120]*Gamma[11/120]/Gamma[1/5]/Gamma[7/40]^2/Gamma[1/8]^2/Gamma[1/60]/Gamma[7/60],
Gamma[41/120]->3^(19/40)/Sqrt[2]/5^(1/8)*Sqrt[G\[Phi]]*Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]*Gamma[1/5]*Gamma[1/40]*Gamma[7/40]/Gamma[1/120]/Gamma[1/20],
Gamma[43/120]->3^(1/4)/5^(5/12)/2^(7/8)*(G\[Psi]+Sqrt[15])^(1/4)*(Sqrt[2]+1)^(3/4)*Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*(Sqrt[5]+Sqrt[3])^(1/4)/
  (Sqrt[3]+1)^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*Gamma[1/3]*Gamma[1/4]*Gamma[3/40]^2/Gamma[1/60]*
  Gamma[1/120]*Gamma[7/120]*Gamma[11/120]*Gamma[2/5]/Gamma[7/40]^2/Gamma[7/60]/Sqrt[Sqrt[10]+3]/Gamma[1/8]^2/Gamma[1/5],
Gamma[47/120]->2^(1/5)*3^(13/40)*5^(1/4)*Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]*Sqrt[Sqrt[10]+3]/Sqrt[Sqrt[2]+1]*
  Gamma[7/40]^2*Gamma[1/8]^2*Gamma[1/5]/Gamma[7/120]/Gamma[1/4]/Gamma[1/40]/Gamma[3/40],
Gamma[49/120]->Sqrt[3]/2^(113/120)/5^(1/24)*Sqrt[G\[Phi]]*(Sqrt[3]+1)^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)* 
  (G\[Psi]+Sqrt[15])^(1/4)*(Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[2]+1)^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*Gamma[1/3]*Gamma[11/120]/Gamma[1/60],
Gamma[53/120]->Sqrt[Pi]*2^(119/120)*Sqrt[5]*(g\[Psi]+Sqrt[15])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*
  (Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[3]+1)^(1/4)/G\[Phi]^(5/4)/(Sqrt[6]+Sqrt[5])^(1/4)/
  (Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[2]+1)^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*Gamma[7/120]/Gamma[7/60],
Gamma[59/120]->Sqrt[Pi]/2^(133/120)/5^(1/4)*Sqrt[G\[Phi]]*(G\[Psi]+Sqrt[15])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[3]+1)^(1/4)*
  (Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[2]+1)^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*Gamma[1/120]/Gamma[1/60],
Gamma[61/120]->Sqrt[Pi]*2^(59/60)*Gamma[1/60]/Gamma[1/120],
Gamma[67/120]->Sqrt[Pi]*2^(53/60)*Gamma[7/60]/Gamma[7/120],
Gamma[71/120]->Pi*2^(31/15)/Sqrt[3]*5^(7/24)/Sqrt[G\[Phi]]/Sqrt[G\[Psi]+Sqrt[15]]/Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]*Gamma[1/60]/Gamma[1/3]/Gamma[11/120],
Gamma[73/120]->Pi*2^(77/40)/3^(13/40)*5^(1/4)/G\[Phi]*(Sqrt[2]+1)^(3/4)*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*
  (Sqrt[3]+1)^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]/Sqrt[Sqrt[10]+3]/(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/
  (Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4)/(g\[Psi]+Sqrt[15])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*
Gamma[1/4]*Gamma[3/40]*Gamma[1/40]*Gamma[7/120]/Gamma[7/40]^2/Gamma[1/8]^2/Gamma[1/5],
Gamma[77/120]->5^(1/6)*Pi*2^(3/2)/3^(1/4)*G\[Phi]^(3/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*Sqrt[Sqrt[10]+3]*Sqrt[Sqrt[g\[Phi]]+Sqrt[3]]*
  Sqrt[Sqrt[g\[Phi]]+Sqrt[5]]/(Sqrt[2]+1)/(G\[Psi]+Sqrt[15])^(1/4)/(g\[Psi]+Sqrt[15])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[10])^(3/4)*Gamma[1/8]^2*
  Gamma[1/5]*Gamma[7/40]^2*Gamma[7/60]*Gamma[1/60]/Gamma[1/3]/Gamma[1/4]/Gamma[3/40]^2/Gamma[2/5]/Gamma[11/120]/Gamma[7/120]/Gamma[1/120],
Gamma[79/120]->Pi*2^(13/8)/3^(19/40)/5^(1/8)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[2]+1)^(1/4)/G\[Phi]^(1/4)/
  (Sqrt[3]+1)^(1/4)/(Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/
  (Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/(G\[Psi]+Sqrt[15])^(1/4)*Gamma[1/20]*Gamma[1/120]/Gamma[1/40]/Gamma[1/5]/Gamma[7/40],
Gamma[83/120]->Pi*2^(15/8)*3^(7/40)*5^(5/12)*(Sqrt[3]+1)^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*
  (Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*Sqrt[Sqrt[10]+3]/
  (Sqrt[2]+1)^(3/4)/(G\[Psi]+Sqrt[15])^(1/4)/(Sqrt[5]+Sqrt[3])^(1/4)/Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*Gamma[1/60]*Gamma[1/5]*Gamma[7/40]^2*
Gamma[1/8]^2*Gamma[7/60]/Gamma[1/3]/Gamma[1/4]/Gamma[11/120]/Gamma[1/120]/Gamma[1/40]/Gamma[2/5]/Gamma[3/40]/Gamma[7/120],
Gamma[89/120]->Pi^(3/2)*2^(299/120)/3^(9/40)*5^(7/24)*Sqrt[Sqrt[10]+3]*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[3]+1)^(1/4)/Sqrt[G\[Phi]]/(Sqrt[2]+1)^(3/4)/(Sqrt[6]+Sqrt[5])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/(G\[Psi]+Sqrt[15])^(1/4)/(Sqrt[5]+Sqrt[3])^(1/4)/Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*Gamma[1/8]^2*Gamma[1/20]*
  Gamma[7/40]*Gamma[1/60]/Gamma[1/40]/Gamma[2/5]/Gamma[3/40]^2/Gamma[1/4]/Gamma[1/3]/Gamma[11/120],
Gamma[91/120]->Pi*4*3^(9/40)*5^(1/8)*Sqrt[Sqrt[10]+3]*Sqrt[Sqrt[G\[Phi]]+Sqrt[5]]*Sqrt[Sqrt[g\[Phi]]+Sqrt[5]]/G\[Phi]^(1/4)/(Sqrt[2]+1)/
  Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*Gamma[1/8]^2*Gamma[1/5]*Gamma[7/40]*Gamma[1/20]/Gamma[1/4]/Gamma[3/40]^2/Gamma[1/40]/Gamma[11/120]/Gamma[2/5],
Gamma[97/120]->Sqrt[Pi]*2^(41/60)*3^(13/40)*5^(1/4)*Sqrt[G\[Phi]]*Sqrt[Sqrt[g\[Phi]]+Sqrt[3]]*Sqrt[Sqrt[G\[Phi]]+Sqrt[5]]*Sqrt[Sqrt[10]+3]*
  (Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(g\[Psi]+Sqrt[15])^(1/4)/Sqrt[Sqrt[2]+1]/(G\[Psi]+Sqrt[15])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*
  Gamma[1/60]*Gamma[7/40]^2*Gamma[1/8]^2*Gamma[1/5]/Gamma[1/120]/Gamma[11/120]/Gamma[1/4]/Gamma[3/40]/Gamma[1/40]/Gamma[7/120],
Gamma[101/120]->Pi*2^(29/30)/3^(1/40)*Gamma[1/20]*Gamma[1/120]/Gamma[1/40]/Gamma[7/40]/Gamma[1/60],
Gamma[103/120]->4*Pi*5^(5/8)*2^(19/120)*3^(1/4)*Sqrt[Sqrt[10]+3]*(Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)/Sqrt[G\[Phi]]*
  (Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*
  (Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[3]+1)^(1/4)/(Sqrt[2]+1)^(3/4)/(G\[Psi]+Sqrt[15])^(1/4)/Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]*
  Gamma[1/8]^2*Gamma[1/60]*Gamma[1/5]*Gamma[7/40]^2/Gamma[1/4]/Gamma[7/120]/Gamma[2/5]/Gamma[3/40]^2/Gamma[11/120]/Gamma[1/120],
Gamma[107/120]->2/15*3^(33/40)*Pi*2^(1/15)*5^(7/8)*Sqrt[Sqrt[2]+1]*Sqrt[Sqrt[g\[Phi]]+Sqrt[10]]/Sqrt[G\[Phi]]/Sqrt[Sqrt[G\[Phi]]+Sqrt[10]]*
  Gamma[1/40]*Gamma[2/5]*Gamma[3/40]*Gamma[1/4]*Gamma[7/120]/Gamma[1/5]/Gamma[7/40]^2/Sqrt[Sqrt[10]+3]/Gamma[1/8]^2/Gamma[7/60],
Gamma[109/120]->Pi*2^(9/8)*(Sqrt[3]+1)^(1/4)*(Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/
  (Sqrt[6]+Sqrt[5])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/
  (G\[Psi]+Sqrt[15])^(1/4)/(Sqrt[2]+1)^(1/4)/Gamma[11/120],
Gamma[113/120]->Pi*2^(3/8)/5^(1/4)*G\[Phi]^(1/4)*(Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)*(g\[Psi]+Sqrt[15])^(1/4)*(Sqrt[2]+1)^(1/4)*
  (Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[3]+1)^(1/4)/(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/Gamma[7/120],
Gamma[119/120]->Pi/2^(1/8)/Sqrt[5]*(Sqrt[3]+1)^(1/4)*Sqrt[G\[Phi]]*(Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*
  (Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[2]+1)^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(G\[Psi]+Sqrt[15])^(1/4)*
  (Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/Gamma[1/120], Gamma[1/2]->Sqrt[Pi] 
};

GammaPsis = { G\[Phi]->5+Sqrt[5], g\[Phi]->5-Sqrt[5], G\[Psi]->Sqrt[5+2Sqrt[5]], g\[Psi]->Sqrt[5-2Sqrt[5]] };

GammaAlgebraicIdentities = Hold[ 
(G\[Psi]+Sqrt[15])*(g\[Psi]+Sqrt[15])==10/G\[Phi]*(G\[Psi]+Sqrt[3])^2,
(G\[Psi]+Sqrt[15])/(g\[Psi]+Sqrt[15])==(g\[Psi]+Sqrt[3])^2/4,
(Sqrt[G\[Phi]]+Sqrt[10])*(Sqrt[g\[Phi]]+Sqrt[10])==(G\[Psi]+Sqrt[5])^2,
(Sqrt[G\[Phi]]+Sqrt[10])/(Sqrt[g\[Phi]]+Sqrt[10])==(g\[Psi]+Sqrt[5])^2/G\[Phi],
(Sqrt[G\[Phi]]+Sqrt[5])/(Sqrt[g\[Phi]]+Sqrt[5])==2*Sqrt[10]/G\[Phi]^(3/2)*(g\[Psi]+Sqrt[10]),
(Sqrt[G\[Phi]]+Sqrt[5])*(Sqrt[g\[Phi]]+Sqrt[5])==G\[Phi]^(3/2)/2/Sqrt[10]*(G\[Psi]+Sqrt[10]),
(Sqrt[G\[Phi]]+Sqrt[3])*(Sqrt[g\[Phi]]+Sqrt[3])==Sqrt[6]*G\[Psi]+3+2*Sqrt[5],
(Sqrt[G\[Phi]]+Sqrt[3])/(Sqrt[g\[Phi]]+Sqrt[3])==Sqrt[6]*G\[Psi]-4-Sqrt[5],
(G\[Psi]+Sqrt[15])*(Sqrt[G\[Phi]]+Sqrt[10])==2^(3/2)*5^(3/2)/G\[Phi]^2/(Sqrt[3]+1)*(Sqrt[5]+Sqrt[3])*(G\[Psi]+Sqrt[3]+2)^2,
(Sqrt[G\[Phi]]+Sqrt[10])/(G\[Psi]+Sqrt[15])==Sqrt[5]/2^(3/2)/G\[Phi]*(Sqrt[3]+1)/(Sqrt[5]+Sqrt[3])*(G\[Psi]-Sqrt[3]+2)^2,
(g\[Psi]+Sqrt[15])*(Sqrt[g\[Phi]]+Sqrt[10])==G\[Phi]^2/2^(7/2)/Sqrt[5]*(Sqrt[3]+1)*(Sqrt[5]+Sqrt[3])*(g\[Psi]-Sqrt[3]+2)^2,
(Sqrt[g\[Phi]]+Sqrt[10])/(g\[Psi]+Sqrt[15])==G\[Phi]/2^(5/2)/Sqrt[5]/(Sqrt[3]+1)/(Sqrt[5]+Sqrt[3])*(g\[Psi]+Sqrt[3]+2)^2,
2*Sqrt[2]+Sqrt[3]+1==2*(Sqrt[2]+1)/(Sqrt[3]+1)*(Sqrt[3]+Sqrt[2]),
2*Sqrt[2]-Sqrt[3]+1==(Sqrt[2]+1)*(Sqrt[3]+1)/(Sqrt[3]+Sqrt[2]),
2*Sqrt[2]+Sqrt[3]-1==(Sqrt[3]+1)/(Sqrt[2]+1)*(Sqrt[3]+Sqrt[2]),
2*Sqrt[2]-Sqrt[3]-1==2/(Sqrt[2]+1)/(Sqrt[3]+1)/(Sqrt[3]+Sqrt[2]),
2*Sqrt[3]+Sqrt[5]+1==2*Sqrt[5]/G\[Phi]*(Sqrt[3]+1)*(Sqrt[5]+Sqrt[3]),
2*Sqrt[3]-Sqrt[5]-1==8*Sqrt[5]/G\[Phi]/(Sqrt[3]+1)/(Sqrt[5]+Sqrt[3]),
2*Sqrt[3]+Sqrt[5]-1==G\[Phi]/Sqrt[5]/(Sqrt[3]+1)*(Sqrt[5]+Sqrt[3]),
2*Sqrt[3]-Sqrt[5]+1==G\[Phi]/Sqrt[5]*(Sqrt[3]+1)/(Sqrt[5]+Sqrt[3]),
1+2*Sqrt[2]+Sqrt[5]==2^(3/2)*5^(1/4)*Sqrt[Sqrt[2]+1]/Sqrt[G\[Phi]]*Sqrt[Sqrt[10]+3],
1+2*Sqrt[2]-Sqrt[5]==Sqrt[2]/5^(1/4)*Sqrt[Sqrt[2]+1]*Sqrt[G\[Phi]]/Sqrt[Sqrt[10]+3],
1-2*Sqrt[2]+Sqrt[5]==2^(3/2)*5^(1/4)/Sqrt[Sqrt[2]+1]/Sqrt[G\[Phi]]/Sqrt[Sqrt[10]+3],
-1+2*Sqrt[2]+Sqrt[5]==Sqrt[2]/5^(1/4)/Sqrt[Sqrt[2]+1]*Sqrt[G\[Phi]]*Sqrt[Sqrt[10]+3],
2+Sqrt[2]+Sqrt[3]+Sqrt[5]==G\[Phi]^(3/2)/2/5^(3/4)*Sqrt[Sqrt[2]+1]*Sqrt[Sqrt[3]+Sqrt[2]]*Sqrt[Sqrt[5]+Sqrt[3]]/Sqrt[Sqrt[6]+Sqrt[5]],
2-Sqrt[2]+Sqrt[3]+Sqrt[5]==G\[Phi]^(3/2)/2/5^(3/4)/Sqrt[Sqrt[2]+1]/Sqrt[Sqrt[3]+Sqrt[2]]*Sqrt[Sqrt[5]+Sqrt[3]]*Sqrt[Sqrt[6]+Sqrt[5]],
2+Sqrt[2]-Sqrt[3]+Sqrt[5]==G\[Phi]^(3/2)/Sqrt[2]/5^(3/4)*Sqrt[Sqrt[2]+1]/Sqrt[Sqrt[3]+Sqrt[2]]/Sqrt[Sqrt[5]+Sqrt[3]]*Sqrt[Sqrt[6]+Sqrt[5]],
2-Sqrt[2]-Sqrt[3]+Sqrt[5]==G\[Phi]^(3/2)/Sqrt[2]/5^(3/4)/Sqrt[Sqrt[2]+1]*Sqrt[Sqrt[3]+Sqrt[2]]/Sqrt[Sqrt[5]+Sqrt[3]]/Sqrt[Sqrt[6]+Sqrt[5]],
2+Sqrt[2]+Sqrt[3]-Sqrt[5]==2^(5/2)*5^(3/4)/G\[Phi]^(3/2)*Sqrt[Sqrt[2]+1]*Sqrt[Sqrt[3]+Sqrt[2]]/Sqrt[Sqrt[5]+Sqrt[3]]*Sqrt[Sqrt[6]+Sqrt[5]],
2-Sqrt[2]+Sqrt[3]-Sqrt[5]==2^(5/2)*5^(3/4)/G\[Phi]^(3/2)/Sqrt[Sqrt[2]+1]/Sqrt[Sqrt[3]+Sqrt[2]]/Sqrt[Sqrt[5]+Sqrt[3]]/Sqrt[Sqrt[6]+Sqrt[5]],
-2-Sqrt[2]+Sqrt[3]+Sqrt[5]==4*5^(3/4)/G\[Phi]^(3/2)*Sqrt[Sqrt[2]+1]/Sqrt[Sqrt[3]+Sqrt[2]]*Sqrt[Sqrt[5]+Sqrt[3]]/Sqrt[Sqrt[6]+Sqrt[5]],
-2+Sqrt[2]+Sqrt[3]+Sqrt[5]==4*5^(3/4)/G\[Phi]^(3/2)/Sqrt[Sqrt[2]+1]*Sqrt[Sqrt[3]+Sqrt[2]]*Sqrt[Sqrt[5]+Sqrt[3]]*Sqrt[Sqrt[6]+Sqrt[5]],
(Sqrt[3]-Sqrt[2])*G\[Psi]+Sqrt[2]+1==G\[Phi]^(7/4)/2^(11/8)/5*(Sqrt[2]+1)^(1/4)*(Sqrt[3]+1)^(3/4)/Sqrt[Sqrt[3]+Sqrt[2]]/
  (Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4)*(g\[Psi]+Sqrt[15])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4),
(Sqrt[3]-Sqrt[2])*G\[Psi]+Sqrt[2]-1==G\[Phi]^2/2^(5/8)/5/(Sqrt[2]+1)^(1/4)/(Sqrt[3]+1)^(3/4)/Sqrt[Sqrt[3]+Sqrt[2]]*
  (Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4)/(g\[Psi]+Sqrt[15])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4),
(Sqrt[3]+Sqrt[2])*G\[Psi]-Sqrt[2]-1==2^(7/8)/5^(1/4)*Sqrt[G\[Phi]]*(Sqrt[2]+1)^(1/4)/(Sqrt[3]+1)^(3/4)*Sqrt[Sqrt[3]+Sqrt[2]]*
  (Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)/(g\[Psi]+Sqrt[15])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4),
(Sqrt[3]+Sqrt[2])*G\[Psi]-Sqrt[2]+1==2^(1/8)/5^(1/4)*G\[Phi]^(1/4)/(Sqrt[2]+1)^(1/4)*(Sqrt[3]+1)^(3/4)*Sqrt[Sqrt[3]+Sqrt[2]]/
  (Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)*(g\[Psi]+Sqrt[15])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*
  (Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4),
(Sqrt[3]+Sqrt[2])*G\[Psi]+Sqrt[2]-1==G\[Phi]^(9/4)/2^(9/8)/5*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)*Sqrt[Sqrt[3]+Sqrt[2]]*
  (Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[3]+1)^(3/4)/(g\[Psi]+Sqrt[15])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)/
  (Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[2]+1)^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4),
(Sqrt[2]-Sqrt[3])*G\[Psi]+Sqrt[2]+1==1/5*2^(3/8)*5^(3/4)*(Sqrt[3]+1)^(3/4)*G\[Phi]^(3/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[2]+1)^(1/4)*
  (Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4)/Sqrt[Sqrt[3]+Sqrt[2]]/
  (Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/(g\[Psi]+Sqrt[15])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[10])^(1/4),
(Sqrt[3]+Sqrt[2])*G\[Psi]+Sqrt[2]+1==1/2^(7/8)/5*(Sqrt[2]+1)^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)*Sqrt[Sqrt[3]+Sqrt[2]]*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*
  G\[Phi]^2*(g\[Psi]+Sqrt[15])^(1/4)*(Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/
  (Sqrt[3]+1)^(3/4)/(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4),
(Sqrt[3]-Sqrt[2])*G\[Psi]-Sqrt[2]+1==2^(5/8)/5^(1/4)*Sqrt[G\[Phi]]*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*(g\[Psi]+Sqrt[15])^(1/4)*
  (Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/((Sqrt[2]+1)^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)*Sqrt[Sqrt[3]+Sqrt[2]]*
  (Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[3]+1)^(3/4)*(Sqrt[g\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)),
(Sqrt[3]+Sqrt[2])*g\[Psi]+Sqrt[2]+1==2^(17/8)*5^(1/4)*Sqrt[Sqrt[3]+Sqrt[2]]*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*
  (Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[2]+1)^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(G\[Psi]+Sqrt[15])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/
  (Sqrt[3]+1)^(3/4)/(Sqrt[5]+Sqrt[3])^(1/4)/G\[Phi]/(Sqrt[6]+Sqrt[5])^(1/4),
(Sqrt[3]+Sqrt[2])*g\[Psi]+Sqrt[2]-1==2^(15/8)*5^(3/4)*(Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[3]+1)^(3/4)*Sqrt[Sqrt[3]+Sqrt[2]]*
  (Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[2]+1)^(1/4)/(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/
  (G\[Psi]+Sqrt[15])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/G\[Phi]^(5/4)/(Sqrt[6]+Sqrt[5])^(1/4),
(Sqrt[2]-Sqrt[3])*g\[Psi]+Sqrt[2]+1==2^(15/8)*Sqrt[5]*(Sqrt[2]+1)^(1/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*
  (Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[3]+1)^(3/4)*(Sqrt[5]+Sqrt[3])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)/G\[Phi]^(5/4)/
  Sqrt[Sqrt[3]+Sqrt[2]]/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/(G\[Psi]+Sqrt[15])^(1/4),
(Sqrt[2]-Sqrt[3])*g\[Psi]+Sqrt[2]-1==2^(17/8)*Sqrt[5]*(G\[Psi]+Sqrt[15])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)/
  (Sqrt[3]+1)^(3/4)/G\[Phi]/(Sqrt[5]+Sqrt[3])^(1/4)/Sqrt[Sqrt[3]+Sqrt[2]]/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/
  (Sqrt[2]+1)^(1/4)/(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[5])^(1/4),
(Sqrt[3]-Sqrt[2])*g\[Psi]+Sqrt[2]-1==2^(27/8)*5^(3/4)*(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*
  (Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/(G\[Psi]+Sqrt[15])^(1/4)/(Sqrt[3]+1)^(3/4)/
  (Sqrt[5]+Sqrt[3])^(1/4)/G\[Phi]^(3/2)/Sqrt[Sqrt[3]+Sqrt[2]]/(Sqrt[2]+1)^(1/4),
(Sqrt[3]+Sqrt[2])*g\[Psi]-Sqrt[2]+1==2^(13/8)*Sqrt[5]*(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[5])^(1/4)*(G\[Psi]+Sqrt[15])^(1/4)*
  (Sqrt[3]+1)^(3/4)*(Sqrt[5]+Sqrt[3])^(1/4)*Sqrt[Sqrt[3]+Sqrt[2]]/(Sqrt[2]+1)^(1/4)/G\[Phi]^(5/4)/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/
  (Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4),
(Sqrt[3]-Sqrt[2])*g\[Psi]+Sqrt[2]+1==2^(13/8)*(Sqrt[2]+1)^(1/4)*(Sqrt[5]+Sqrt[3])^(1/4)*(G\[Psi]+Sqrt[15])^(1/4)*(Sqrt[3]+1)^(3/4)*5^(3/4)*
  (Sqrt[G\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)*(Sqrt[6]+Sqrt[5])^(1/4)/G\[Phi]^(5/4)/(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/
  (Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/Sqrt[Sqrt[3]+Sqrt[2]],
-(Sqrt[3]+Sqrt[2])*g\[Psi]+Sqrt[2]+1==5*2^(27/8)*Sqrt[Sqrt[3]+Sqrt[2]]*(Sqrt[2]+1)^(1/4)/(Sqrt[3]+1)^(3/4)/G\[Phi]^(3/2)/
  (Sqrt[5]+Sqrt[3])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[g\[Phi]]+Sqrt[3])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[5])^(1/4)/
  (Sqrt[g\[Phi]]+Sqrt[5])^(1/4)/(Sqrt[6]+Sqrt[5])^(1/4)/(Sqrt[G\[Phi]]+Sqrt[10])^(1/4)/(G\[Psi]+Sqrt[15])^(1/4)
];

Gamma2EllipticK = {
Gamma[1/3]->2^(7/9)*Pi^(1/3)/3^(1/12)*EllipticK[(2-Sqrt[3])/4]^(1/3),
Gamma[1/4]->2*Pi^(1/4)*EllipticK[1/2]^(1/2),
Gamma[1/8]->Pi^(1/8)*2^(17/8)*EllipticK[1/2]^(1/4)*EllipticK[3-2*Sqrt[2]]^(1/2),
Gamma[1/15]->Pi^(1/6)*2^(8/9)*3^(29/60)*5^(13/24)/Sqrt[5+Sqrt[5]]*Sqrt[Sqrt[5+2*Sqrt[5]]+Sqrt[3]]*Gamma[1/5]^(1/2)/Gamma[2/5]^(1/2)
*EllipticK[(2-Sqrt[3])/4]^(1/6)*EllipticK[(7-3*Sqrt[5])*(4-Sqrt[15])*(7-4*Sqrt[3])/32]^(1/2),
Gamma[1/20]->2^(9/40)*5^(1/8)/Pi^(1/4)*(5+Sqrt[5])^(5/8)*Sqrt[1+Sqrt[5-2*Sqrt[5]]]
*Gamma[1/5]^(1/2)*Gamma[2/5]^(1/2)*EllipticK[1/2-Sqrt[Sqrt[5]-2]]^(1/2),
Gamma[1/24]->2^(49/18)*3^(25/48)*Pi^(1/24)*Sqrt[Sqrt[2]+1]/(Sqrt[3]+1)^(1/4)
*EllipticK[1/2]^(1/4)*EllipticK[(2-Sqrt[3])/4]^(1/6)*EllipticK[(7-4*Sqrt[3])*(5-2*Sqrt[6])]^(1/2)
};

End[]

EndPackage[]
