FUNCTION :  findtype8 - find type 8 identities
                

CALLING SEQUENCE :  findtype8(T)
                    

PARAMETERS :   T - positive integer  
                  -            

GLOBAL VARIABLES : 
        xprint,NEWJACID,RJID,SYMJID,
        TT1,TT2,PROVEDFL1,EBL

SYNOPSIS :   
        Before running the functions G,H,GM,HM,GE,HE must be defined.   

        findtype8(T) cycles through symbolic expressions
        
           _G(1)^a_H(a) + c _H(1)^a_G(a)
        
        where  2 ≤ a ≤  c in {-1,1},  

        using CHECKRAMIDF to check whether the expression corresponds to a 
        likely eta-product.
        If proveit is true then provemodfuncidBATCH (from theataids 
        package) is used to prove it.  The procedure also returns a list 
        of [a,c] which give identities.

EXAMPLES :   

>  with(qseries):
>  with(thetaids):
>  with(ramarobinsids):
>  xprint:=false: proveit:=true:
>  G:=j->1/GetaL([1],5,j):H:=j->1/GetaL([2],5,j):
>  GM:=j->1/MGetaL([1],5,j): HM:=j->1/MGetaL([2],5,j):
>  GE:=j->-GetaLEXP([1],5,j):HE:=j->-GetaLEXP([2],5,j):
>  findtype8(12);
*** There were NO errors.  Each term was modular function on
    Gamma1(15). Also -mintotord=4. To prove the identity
    we need to  check up to O(q^(6)).
    To be on the safe side we check up to O(q^(34)).
*** The identity below is PROVED!
[3, -1]
                                                     3        
       3              3                 3 eta(15 tau)         
  _G(1)  _H(3) - _H(1)  _G(3) = ------------------------------
                                eta(5 tau) eta(3 tau) eta(tau)
"n=", 10
WARNING: There were 2 ebasethreshold problems.
         See the global array EBL.
                           [[3, -1]]


DISCUSSION :
    For G,H defined 
    G(1)^3 H(3) - H(1)^3 G(3), is an eta-product and the 
    identity is proved.        

SEE ALSO :  

findtype6, findtype2,
findtype3, findtype6,
findtype6, findtype6,
findtype7, findtype8,
findtype9, findtype60