FUNCTION :  findtype7 - find type 7 identities
                

CALLING SEQUENCE :  findtype7(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.   

        findtype7(T) cycles through symbolic expressions
        
           _GM(a)_G(b) + c _HM(a)_H(b)
        
        where  2 ≤ n ≤ T, ab=n, (a,b)=1,  b < a, c in {-1,1},  

        (*)    GE(a) + GE(b) - (HE(a) + HE(b))  in Z

        and least one of a, b is even,
        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,b,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):
>  findtype7(12);
*** There were NO errors.  Each term was modular function on
    Gamma1(180). Also -mintotord=288. To prove the identity
    we need to  check up to O(q^(290)).
    To be on the safe side we check up to O(q^(648)).
*** The identity below is PROVED!
[9, 1, -1]
         _GM(1) _G(9) - _HM(1) _H(9) = 

                           2                           
                eta(18 tau)  eta(12 tau) eta(tau)      
           --------------------------------------------
           eta(36 tau) eta(9 tau) eta(6 tau) eta(2 tau)
"n=", 10
                          [[1, 9, -1]]


DISCUSSION :
    For G,H defined 
    G*(1) G(9) - H*(1) H (9), is an eta-product and the 
    identity is proved.        

SEE ALSO :  

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