FUNCTION : ETA[provemodfuncGAMMA0UpETAidBATCH] - proving U[p] eta-combo identity
CALLING SEQUENCE : provemodfuncGAMMA0UpETAidBATCH()
provemodfuncGAMMA0UpETAidBATCH(EP,p,etacombo,N)
PARAMETERS :
EP - one eta-product
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 :
qcheck,modfunccheck, totcheck, _ORDS,jptmp,jpqd,eptmp,gltmp
etaPRODL,GPL,COFS,conpres,CONTERMS,mintottmp,consL,MFLB
xprint,_CUSPS,UpORDL,_ORDS2,noprint
SYNOPSIS :
This a BATCH version of provemodfuncGAMMA0UpETAid
This function PROVES the id U[p](EP) = etacombo
EXAMPLES :
> with(qseries):
> with(ETA):
> provemodfuncGAMMA0UpETAidBATCH();
-------------------------------------------------------------
provemodfuncGAMMA0UpETAidBATCH(EP,p,etacombo,N)
This a BATCH version of provemodfuncGAMMA0UpETAid
EP = one eta-product
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](EP) = etacombo
global vars (can be used for error-checking):
qcheck, modfunccheck, totcheck, _ORDS, jptmp, jpqd, eptmp,
gltmp, EPRODL, GETAL, COFS, conpres, CONTERMS, mintottmp
-------------------------------------------------------------
> gpg:=[100, -3, 50, 5, 25, -2, 10, -8, 5, 4, 4, 3, 2, 3, 1, -2]:
> epg:=gp2etaprod(gpg);
5 4 3 3
eta(50 tau) eta(5 tau) eta(4 tau) eta(2 tau)
epg := -------------------------------------------------
3 2 8 2
eta(100 tau) eta(25 tau) eta(10 tau) eta(tau)
> gpf1:=[10, 8, 5, -4, 2, -8, 1, 4]:
> epf1:=gp2etaprod(gpf1);
8 4
eta(10 tau) eta(tau)
epf1 := -----------------------
4 8
eta(5 tau) eta(2 tau)
> gpf2:=[20, -3, 10, 5, 5, -2, 4, -1, 2, -1, 1, 2]:
> epf2:=gp2etaprod(gpf2);
5 2
eta(10 tau) eta(tau)
epf2 := ----------------------------------------------
3 2
eta(20 tau) eta(5 tau) eta(4 tau) eta(2 tau)
> etacombo:=5*epf1 + 2*epf2:
> noprint:=false:
> provemodfuncGAMMA0UpETAidBATCH(epg,5,etacombo,20);
*** There were NO errors.
*** o EP 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).
memory used=4.2MB, alloc=40.3MB, time=0.09
We find that LHS - RHS is
43
O(q )
43
[1, -3, O(q )]
> noprint:=true:
> provemodfuncGAMMA0UpETAidBATCH(epg,5,etacombo,20);
43
[1, -3, O(q )]
DISCUSSION :
SEE ALSO :