FUNCTION :  ETA[provemodfuncGAMMA0UpETAid] -  prove U[p] eta-product identity

CALLING SEQUENCE :  provemodfuncGAMMA0UpETAid()
                    provemodfuncGAMMA0UpETAid(gcombo,p,etacombo,N)

PARAMETERS :    
  gcombo - sum of modular functions on Gamma[0](p*N)
       p - prime                                             
etacombo - sum of modular functions on Gamma[0](N)           
           Each term in the sum is a eta-quotient to base N. 
       N - Positive integer multiple of p                    
                                                             
GLOBAL VARIABLES : 

SYNOPSIS :   
   This function PROVES the id U[p](EP) = etacombo           
   global vars (can be used for error-checking):             
   qcheck, modfunccheck, totcheck, _ORDS, jptmp, jpqd, eptmp,
   gltmp, EPRODL, GETAL, COFS, conpres, CONTERMS, mintottmp
   

EXAMPLES :   

> with(ETA):
> provemodfuncGAMMA0UpETAid();
-------------------------------------------------------------
provemodfuncGAMMA0UpETAid(gcombo,p,etacombo,N)
  gcombo = sum of modular functions on Gamma[0](p*N)
       p = prime
etacombo = sum of modular functions on Gamma[0](N)
           Each term in the sum is a eta-quotient to base N. 
       N = Positive integer multiple of p
                                                             
   This function PROVES the id U[p](gcombo) = etacombo
   global vars (can be used for error-checking):
   qcheck, modfunccheck, totcheck, _ORDS, jptmp, jpqd, eptmp,
   gltmp, EPRODL, GETAL, COFS, conpres, CONTERMS, mintottmp  

-------------------------------------------------------------

> with(qseries):
> with(ETA):
> gpY1a := [50, 2, 25, -4, 20, -1, 10, -2, 5, 3, 4, -3, 2, 8, 1, -3]:
> gpY1b := [50, 2, 25, -4, 20, 1, 10, -5, 5, 4, 4, 3, 2, -1]:
> gpZ1a := [1, 3, 10, 10, 2, -2, 5, -7, 20, -3, 4, -1]:
> gpZ1b := [1, 4, 10, 1, 2, -5, 5, -4, 20, 3, 4, 1]:
> epY1a:=gp2etaprod(gpY1a):
> epY1b:=gp2etaprod(gpY1b):
> epZ1a:=gp2etaprod(gpZ1a):
> epZ1b:=gp2etaprod(gpZ1b):
> gcombo:=epY1a-4*epY1b;
> etacombo:=-epZ1a-4*epZ1b;
> provemodfuncGAMMA0UpETAid(gcombo,5,etacombo,20);

*** There were NO errors. 
*** o each EP in gcombo is an MF on Gamma[0](100)
*** o Each term in the etacombo is a  modular function on
      Gamma0(20).
*** o We also checked that the total order of
      each term etacombo was zero.
*** To prove the identity U[5](EP)=etacombo we need to show
    that v[oo](ID) > 3    This means checking up to q^(4).
Do you want to prove the identity? (yes/no)
yes                                                                               
You entered yes.
We verify the identity to O(q^(43)).
We find that LHS - RHS is 
                                         43
                                      O(q  )

RESULT: The identity holds to O(q^(43)).
CONCLUSION: This proves the identity since we had only
            to show that v[oo](ID) > 3.

DISCUSSION :

SEE ALSO :