#######################################################################
# Ancillary file for arXiv:2308.08199 [hep-th],
# "An Eight Loop Amplitude via Antipodal Duality",
# by Lance Dixon and Yu-Ting (Andy) Liu.
#######################################################################
# This file describes the hexagon function space that is obtained
# when normalizing the amplitude using the new cosmic normalization "rho"
# (sometimes called "EZ" normalization).
# It is a bit smaller than the space obtain using the old cosmic normalization
# "rho_old" (sometimes called "EX" normalization).
# The spaces start to be smaller starting at weight 6 even, and weight 7 odd.
# Both reductions are associated with locking the coefficients of
# functions of the form
#
#  z4 * [functions with weight lower by 4]
#
# in linear combinations with top-weight functions.
# Here z4 = Zeta[4] = Pi^4/90.
#
##############################################################################
#
# The weight 6 odd space, with dimension 13, is good as is;
# nothing needs to be removed,
# and indeed, there is no function of the form   z4 * [weight 2]
# in the odd sector.
#
odd6Zlist := [ YO(6,1), YO(6,2), YO(6,3), YO(6,4), YO(6,5), YO(6,6),
YO(6,7), YO(6,8), YO(6,9), YO(6,10), YO(6,11), YO(6,12), YO(6,13) ] :
#
##################
# The weight 6 even space from {6,1,1,1,1,1,1,1,1} coproducts of
# the 8 loop MHV amplitude EZMHV8 (as well as lower-loop amplitudes)
# has dimension 89.
# It has z4 * YE(2,{4,5,6}) = YE(6,{89,90,91}) locked to the
# first 88 weight 6 even functions
# (and then there is z6 = YE(6,92), which is also independent):
#
even6Zlist := [
YE(6,1)-12*YE(6,89), YE(6,2)-12*YE(6,90), YE(6,3)-12*YE(6,91),
YE(6,4), YE(6,5), YE(6,6), YE(6,7), YE(6,8), YE(6,9),
YE(6,10), YE(6,11), YE(6,12), YE(6,13), YE(6,14), YE(6,15),
YE(6,16), YE(6,17), YE(6,18), YE(6,19), YE(6,20), YE(6,21),
YE(6,22), YE(6,23), YE(6,24), YE(6,25)+12*YE(6,89), YE(6,26), YE(6,27),
YE(6,28), YE(6,29)+12*YE(6,90), YE(6,30)+12*YE(6,91),
YE(6,31)+9*YE(6,89), YE(6,32), YE(6,33), YE(6,34), YE(6,35), YE(6,36),
YE(6,37), YE(6,38), YE(6,39)+9*YE(6,90), YE(6,40), YE(6,41), YE(6,42),
YE(6,43), YE(6,44), YE(6,45), YE(6,46), YE(6,47), YE(6,48),
YE(6,49), YE(6,50), YE(6,51)+9*YE(6,91),
YE(6,52), YE(6,53)+12*YE(6,89), YE(6,54)+12*YE(6,90),
YE(6,55)+12*YE(6,91), YE(6,56), YE(6,57),
YE(6,58), YE(6,59)-12*YE(6,89), YE(6,60)-12*YE(6,90),
YE(6,61)-12*YE(6,91), YE(6,62), YE(6,63),
YE(6,64), YE(6,65)-6*YE(6,90)-6*YE(6,91), YE(6,66)-6*YE(6,89)-6*YE(6,90),
YE(6,67)+6*YE(6,90), YE(6,68)-6*YE(6,89), YE(6,69)-6*YE(6,89)-6*YE(6,91),
YE(6,70)-6*YE(6,90), YE(6,71)-6*YE(6,91),
   YE(6,72)-6*YE(6,89)-6*YE(6,90)+18*YE(6,91),
YE(6,73)-12*YE(6,89)-30*YE(6,90), YE(6,74)-6*YE(6,89)-6*YE(6,90),
   YE(6,75)-12*YE(6,91),
YE(6,76)+30*YE(6,89)+12*YE(6,90), YE(6,77)+12*YE(6,89)+12*YE(6,90)+21*YE(6,91),
   YE(6,78)+21*YE(6,89)+12*YE(6,90)+12*YE(6,91),
YE(6,79)-6*YE(6,89)+9*YE(6,90)-6*YE(6,91), YE(6,80)+6*YE(6,89)+6*YE(6,90),
   YE(6,81)-6*YE(6,89)-12*YE(6,90),
YE(6,82)+6*YE(6,90)+6*YE(6,91), YE(6,83)+12*YE(6,89)+6*YE(6,90)+18*YE(6,91),
   YE(6,84)-12*YE(6,91),
YE(6,85)-12*YE(6,89)+6*YE(6,90)-9*YE(6,91), YE(6,86), YE(6,87),
YE(6,88),
YE(6,92) ] :
#
# The coefficients are always a multiple of 3.
#
# These 102 weight 6 functions span the weight 6 coproducts of
# EZMHV through 8 loops, and {EZ,EZt} through 6 loops.
#
# Even and odd together (in maple):
#
all6Zlist := [ op(even6Zlist), op(odd6Zlist) ]:
#
###############################################################
# The weight 7 odd space from {7,1,1,1,1,1,1,1} coproducts of
# EZMHV8 (and lower loops) has dimension 29.
# It has z4 * YO(3,1) = YO(7,30) locked to the other 29:
#
odd7Zlist := [ YO(7,1)+9*YO(7,30), YO(7,2)-12*YO(7,30), YO(7,3)-12*YO(7,30),
YO(7,4)-12*YO(7,30), YO(7,5)-9*YO(7,30), YO(7,6)-9*YO(7,30), YO(7,7)+9*YO(7,30),
YO(7,8)+48*YO(7,30), YO(7,9)-9*YO(7,30), YO(7,10)+3*YO(7,30),
YO(7,11)-39*YO(7,30), YO(7,12)-18*YO(7,30), YO(7,13)-51*YO(7,30),
YO(7,14)-6*YO(7,30), YO(7,15)+9*YO(7,30), YO(7,16)-30*YO(7,30),
YO(7,17)+12*YO(7,30), YO(7,18)-21*YO(7,30), YO(7,19)+27*YO(7,30),
YO(7,20)+15*YO(7,30), YO(7,21)+6*YO(7,30), YO(7,22)+3*YO(7,30),
YO(7,23)-27*YO(7,30), YO(7,24)-6*YO(7,30), YO(7,25)+24*YO(7,30),
YO(7,26)-30*YO(7,30), YO(7,27)+12*YO(7,30), YO(7,28)-33*YO(7,30),
YO(7,29)-30*YO(7,30) ] :
#
# The coefficients are always a multiple of 3.
#
###############################################################
# The weight 7 even space from {7,1,1,1,1,1,1,1} coproducts of
# EZMHV8 (and lower loops)
# has dimension 161.  It has 9 functions of the form
#    z4 * YE(3,{...}) = YE(7,{156,157,158,162,163,164,165,166,167})
# which are locked to the first 155,
# while YE(7,{159,160,161,168,169,170}) are not locked.
#
even7Zlist := [
YE(7,1)-12*YE(7,156), YE(7,2)-12*YE(7,157), YE(7,3)-12*YE(7,158), 
YE(7,4), YE(7,5), YE(7,6), YE(7,7)+12*YE(7,156), YE(7,8)+12*YE(7,157), YE(7,9)
+12*YE(7,158), YE(7,10), YE(7,11), YE(7,12), YE(7,13), YE(7,14), YE(7,15), YE(
7,16), YE(7,17), YE(7,18), YE(7,19), YE(7,20), YE(7,21), YE(7,22), YE(7,23), 
YE(7,24), YE(7,25)-12*YE(7,156), YE(7,26)-12*YE(7,157), YE(7,27)-12*YE(7,158),
YE(7,28), YE(7,29)-12*YE(7,163), YE(7,30)-12*YE(7,165), YE(7,31)-12*YE(7,167),
YE(7,32)+12*YE(7,156), YE(7,33)+12*YE(7,157), YE(7,34), YE(7,35), YE(7,36)+12*
YE(7,163), YE(7,37)+9*YE(7,156), YE(7,38), YE(7,39)+12*YE(7,163), YE(7,40)-12*
YE(7,163), YE(7,41), YE(7,42)+12*YE(7,167), YE(7,43)+9*YE(7,163), YE(7,44)+9*
YE(7,162), YE(7,45), YE(7,46), YE(7,47), YE(7,48)+12*YE(7,156), YE(7,49), YE(7
,50)-12*YE(7,156), YE(7,51), YE(7,52)+12*YE(7,165), YE(7,53), YE(7,54), YE(7,
55), YE(7,56), YE(7,57)+9*YE(7,157), YE(7,58)+12*YE(7,165), YE(7,59)-12*YE(7,
165), YE(7,60), YE(7,61), YE(7,62)+9*YE(7,165), YE(7,63)+9*YE(7,164), YE(7,64)
, YE(7,65)+12*YE(7,157), YE(7,66), YE(7,67)-12*YE(7,157), YE(7,68), YE(7,69),
YE(7,70)+9*YE(7,158), YE(7,71), YE(7,72)+12*YE(7,167), YE(7,73), YE(7,74), YE(
7,75)+9*YE(7,167), YE(7,76)+9*YE(7,166), YE(7,77), YE(7,78), YE(7,79)+12*YE(7,
158), YE(7,80)+12*YE(7,162)-12*YE(7,163), YE(7,81)+12*YE(7,164)-12*YE(7,165),
YE(7,82)+12*YE(7,166)-12*YE(7,167), YE(7,83), YE(7,84), YE(7,85), YE(7,86), YE
(7,87), YE(7,88), YE(7,89), YE(7,90), YE(7,91), YE(7,92), YE(7,93), YE(7,94),
YE(7,95), YE(7,96), YE(7,97)-12*YE(7,162), YE(7,98)-12*YE(7,164), YE(7,99)-12*
YE(7,166), YE(7,100)-6*YE(7,165)-6*YE(7,167), YE(7,101)-6*YE(7,163)-6*YE(7,164
)+18*YE(7,165), YE(7,102)+6*YE(7,164)-6*YE(7,165), YE(7,103)+12*YE(7,163)+6*YE
(7,164)-6*YE(7,165)-6*YE(7,166)+6*YE(7,167), YE(7,104)-6*YE(7,167), YE(7,105)+
3*YE(7,164)+6*YE(7,165), YE(7,106)-6*YE(7,162)+18*YE(7,163)-6*YE(7,166)+6*YE(7
,167), YE(7,107)+6*YE(7,165), YE(7,108)-6*YE(7,162)+6*YE(7,163)+6*YE(7,165)+12
*YE(7,167), YE(7,109)+6*YE(7,162)+12*YE(7,163)-6*YE(7,165)+12*YE(7,167), YE(7,
110)+24*YE(7,163)-18*YE(7,165)+18*YE(7,167), YE(7,111)+12*YE(7,162)-15*YE(7,
163)+6*YE(7,164)-12*YE(7,165)+12*YE(7,166)-27*YE(7,167), YE(7,112)-9*YE(7,157)
+3*YE(7,162)-21*YE(7,163)+9*YE(7,165)+3*YE(7,167), YE(7,113)+12*YE(7,162)-3*YE
(7,163)-12*YE(7,164)+27*YE(7,165)-6*YE(7,166)+12*YE(7,167), YE(7,114)-9*YE(7,
163)+3*YE(7,164)-12*YE(7,165)+9*YE(7,166)-27*YE(7,167), YE(7,115)-6*YE(7,162)+
12*YE(7,163)+15*YE(7,164)-6*YE(7,165)+12*YE(7,166)+9*YE(7,167), YE(7,116)+9*YE
(7,156)+6*YE(7,162)-18*YE(7,163)-18*YE(7,165)+12*YE(7,166)-12*YE(7,167), YE(7,
117)+9*YE(7,156)+9*YE(7,157)+9*YE(7,158)-3*YE(7,165)-9*YE(7,167), YE(7,118)+18
*YE(7,156)+9*YE(7,157)+9*YE(7,158)-12*YE(7,163)+6*YE(7,164)-9*YE(7,165)-3*YE(7
,167), YE(7,119)+18*YE(7,158)-18*YE(7,163)-9*YE(7,166)+15*YE(7,167), YE(7,120)
-9*YE(7,156)-9*YE(7,157)-9*YE(7,162)+12*YE(7,163)-15*YE(7,164)+45*YE(7,165)+9*
YE(7,166)-24*YE(7,167), YE(7,121)+12*YE(7,157)+36*YE(7,158)-21*YE(7,164)+39*YE
(7,165)-12*YE(7,166)+36*YE(7,167), YE(7,122)+24*YE(7,156)+36*YE(7,158)-6*YE(7,
162)+15*YE(7,163)-12*YE(7,164)+12*YE(7,165)-9*YE(7,166)+30*YE(7,167), YE(7,123
)+12*YE(7,156)+21*YE(7,157)+9*YE(7,158)+12*YE(7,162)-6*YE(7,164)-6*YE(7,165)+6
*YE(7,166)-6*YE(7,167), YE(7,124)+12*YE(7,156)+9*YE(7,157)+33*YE(7,158)+12*YE(
7,162)+12*YE(7,163)-30*YE(7,165), YE(7,125)+9*YE(7,156)+12*YE(7,157)-21*YE(7,
158)-12*YE(7,162)+12*YE(7,163)+18*YE(7,164)-27*YE(7,165)+18*YE(7,166)-15*YE(7,
167), YE(7,126)+21*YE(7,156)+12*YE(7,157)+12*YE(7,162)-3*YE(7,163)-21*YE(7,164
)+54*YE(7,165)+12*YE(7,166)-42*YE(7,167), YE(7,127)+9*YE(7,156)+12*YE(7,157)+
12*YE(7,158)+6*YE(7,162)-18*YE(7,163)-18*YE(7,165)+12*YE(7,166)-12*YE(7,167),
YE(7,128)+12*YE(7,156)+15*YE(7,158)-9*YE(7,162)+30*YE(7,163)+12*YE(7,164)-6*YE
(7,165)-9*YE(7,166)+36*YE(7,167), YE(7,129)+21*YE(7,158)-12*YE(7,163)+3*YE(7,
164)+6*YE(7,165)+6*YE(7,166), YE(7,130)+12*YE(7,156)-6*YE(7,162)+6*YE(7,163)-6
*YE(7,166)+6*YE(7,167), YE(7,131)+3*YE(7,156)+3*YE(7,157)+18*YE(7,163)+6*YE(7,
164)+6*YE(7,165), YE(7,132)+21*YE(7,156)-12*YE(7,158)+9*YE(7,162)+12*YE(7,163)
-6*YE(7,166)+6*YE(7,167), YE(7,133)-21*YE(7,156)+12*YE(7,162)+12*YE(7,163)+6*
YE(7,164)-6*YE(7,165), YE(7,134)+21*YE(7,156)+3*YE(7,157)-9*YE(7,158)+15*YE(7,
162)-6*YE(7,163)+12*YE(7,165)+12*YE(7,167), YE(7,135)-33*YE(7,157)+24*YE(7,158
)-21*YE(7,162)+9*YE(7,163)-6*YE(7,164)+18*YE(7,165)+6*YE(7,166)-18*YE(7,167),
YE(7,136)+12*YE(7,156)+12*YE(7,157)-12*YE(7,163)-12*YE(7,166)+24*YE(7,167), YE
(7,137)-12*YE(7,158)+9*YE(7,162)-12*YE(7,164)+6*YE(7,165)-18*YE(7,166)+24*YE(7
,167), YE(7,138)-9*YE(7,156)-30*YE(7,157)+9*YE(7,162)-18*YE(7,163)+12*YE(7,164
)-6*YE(7,165)-21*YE(7,167), YE(7,139)-12*YE(7,156)-12*YE(7,163)+12*YE(7,164)-\
24*YE(7,165), YE(7,140)+12*YE(7,156)+12*YE(7,158)-18*YE(7,162)+24*YE(7,163)-12
*YE(7,164)+24*YE(7,165)-24*YE(7,166)+48*YE(7,167), YE(7,141)+21*YE(7,156)+39*
YE(7,157)+24*YE(7,163)+12*YE(7,164)-33*YE(7,165), YE(7,142)+12*YE(7,157)-21*YE
(7,158)+6*YE(7,162)-6*YE(7,163)-3*YE(7,164)-30*YE(7,165)+12*YE(7,166)-6*YE(7,
167), YE(7,143)-3*YE(7,156)+24*YE(7,157)-3*YE(7,162)+18*YE(7,163)-12*YE(7,164)
+30*YE(7,166)-18*YE(7,167), YE(7,144)-15*YE(7,156)+18*YE(7,157)+6*YE(7,158)-21
*YE(7,162)+12*YE(7,163)-21*YE(7,164)+24*YE(7,165)-21*YE(7,166)-12*YE(7,167), 
YE(7,145)+27*YE(7,157)+9*YE(7,158)+12*YE(7,165), YE(7,146)-3*YE(7,157)+21*YE(7
,158)+18*YE(7,162)-12*YE(7,163)-6*YE(7,164)+18*YE(7,165), YE(7,147)+12*YE(7,
156)+18*YE(7,157)+9*YE(7,158)-24*YE(7,163)+27*YE(7,164)-39*YE(7,165)+6*YE(7,
167), YE(7,148)+15*YE(7,156)-3*YE(7,157)+12*YE(7,163), YE(7,149)+9*YE(7,156)+3
*YE(7,157)+12*YE(7,158)+18*YE(7,162)-6*YE(7,163)+15*YE(7,164)-36*YE(7,165)+18*
YE(7,166)-6*YE(7,167), YE(7,150)+42*YE(7,156)-12*YE(7,157)-9*YE(7,158)+12*YE(7
,162)-36*YE(7,163)+18*YE(7,164)-42*YE(7,165)+12*YE(7,166)-18*YE(7,167), YE(7,
151)-21*YE(7,156)-39*YE(7,157)-6*YE(7,158)+6*YE(7,162)-27*YE(7,163)+12*YE(7,
164)-3*YE(7,165)-27*YE(7,166)+27*YE(7,167), YE(7,152)+12*YE(7,156)+30*YE(7,157
)+15*YE(7,158)-12*YE(7,162)+24*YE(7,163)+12*YE(7,165), YE(7,153)-9*YE(7,156)-3
*YE(7,158)-12*YE(7,163), YE(7,154)+15*YE(7,156)+36*YE(7,157)-12*YE(7,158)+21*
YE(7,162)+12*YE(7,163)-12*YE(7,164)+36*YE(7,165)-6*YE(7,166)+18*YE(7,167), YE(
7,155)+3*YE(7,157)-60*YE(7,158)+6*YE(7,162)+6*YE(7,163)+18*YE(7,164)-30*YE(7,
165)-6*YE(7,166),
# these 6 are already free:
YE(7,159), YE(7,160), YE(7,161),
YE(7,168), YE(7,169), YE(7,170) ] :
#
# The coefficients are always a multiple of 3.
#
# Even and odd together (in maple):
#
all7Zlist := [ op(even7Zlist), op(odd7Zlist) ]:
#
###############################################################
# The weight 8 odd space from {8,1,1,1,1,1,1} coproducts of
# EZMHV8 (and lower loops) has dimension 57.
# It has the two functions
# z4 * YO(4,{1,2}) = YO(8,{58,59}) locked to the other 57:
#
odd8Zlist := [
YO(8,1)-12*YO(8,59), YO(8,2)+12*YO(8,58)-12*YO(8,59), YO(8,3)-12*YO
(8,58)+12*YO(8,59), YO(8,4)-12*YO(8,59), YO(8,5)-12*YO(8,59), YO(8,6)-12*YO(8,
58), YO(8,7)+9*YO(8,58), YO(8,8)-18*YO(8,58)+12*YO(8,59), YO(8,9)+9*YO(8,59),
YO(8,10)+6*YO(8,58)-18*YO(8,59), YO(8,11)-12*YO(8,59), YO(8,12)-18*YO(8,58)+9*
YO(8,59), YO(8,13)-21*YO(8,59), YO(8,14)+9*YO(8,58), YO(8,15)-21*YO(8,58)+21*
YO(8,59), YO(8,16)+3*YO(8,58)-9*YO(8,59), YO(8,17)+21*YO(8,59), YO(8,18)+3*YO(
8,58)-18*YO(8,59), YO(8,19)-9*YO(8,58)+18*YO(8,59), YO(8,20)-12*YO(8,58), YO(8
,21)-24*YO(8,58)+12*YO(8,59), YO(8,22)-21*YO(8,58)-9*YO(8,59), YO(8,23)+21*YO(
8,58)-48*YO(8,59), YO(8,24)-33*YO(8,58)+6*YO(8,59), YO(8,25)-15*YO(8,58)+6*YO(
8,59), YO(8,26)-27*YO(8,58)+39*YO(8,59), YO(8,27)-39*YO(8,58)-18*YO(8,59), YO(
8,28)-9*YO(8,58)+18*YO(8,59), YO(8,29)+6*YO(8,58)+24*YO(8,59), YO(8,30)+12*YO(
8,58), YO(8,31)+6*YO(8,58)-27*YO(8,59), YO(8,32)-33*YO(8,59), YO(8,33)-9*YO(8,
58)-18*YO(8,59), YO(8,34)+18*YO(8,58)-57*YO(8,59), YO(8,35)-6*YO(8,58)-3*YO(8,
59), YO(8,36)+21*YO(8,58), YO(8,37)-6*YO(8,58)-48*YO(8,59), YO(8,38)+15*YO(8,
58)+36*YO(8,59), YO(8,39)+51*YO(8,58)-3*YO(8,59), YO(8,40)+12*YO(8,58)-9*YO(8,
59), YO(8,41)+21*YO(8,58)-3*YO(8,59), YO(8,42)+18*YO(8,59), YO(8,43)-9*YO(8,58
)-30*YO(8,59), YO(8,44)+33*YO(8,58)-27*YO(8,59), YO(8,45)+6*YO(8,59), YO(8,46)
-9*YO(8,58)+18*YO(8,59), YO(8,47)-60*YO(8,58)+3*YO(8,59), YO(8,48)+63*YO(8,58)
+36*YO(8,59), YO(8,49)-48*YO(8,58)+45*YO(8,59), YO(8,50)+6*YO(8,58)+12*YO(8,59
), YO(8,51)+18*YO(8,58)-18*YO(8,59), YO(8,52)+45*YO(8,58)-33*YO(8,59), YO(8,53
)+30*YO(8,58)+15*YO(8,59), YO(8,54)-12*YO(8,58), YO(8,55)-18*YO(8,58)+6*YO(8,
59), YO(8,56)-48*YO(8,58), YO(8,57)-42*YO(8,58)-63*YO(8,59) ] :
#
# The coefficients are always a multiple of 3.
#
###################
# The weight 8 even space is not officially saturated, although it only
# went up in dimension from 280 to 286 with the addition of 8 loop MHV.
#
###############################################################
# The weight 9 odd space from {9,1,1,1,1,1} coproducts of
# EZMHV8 (and lower loops) has dimension 113.
# It has   z4 * YO(5,{...}) = YO(9,{114,...,119})
#    and   z6 * YO(3,1) = YO(9,120)
# locked to the other 113:
#
odd9Zlist := [
YO(9,1)+9*YO(9,114)-YO(9,120), YO(9,2)+30*YO(9,114)+18*YO(9,116)+80
*YO(9,120), YO(9,3)-3*YO(9,116)+YO(9,120), YO(9,4)+12*YO(9,114)-3*YO(9,115)-85
*YO(9,120), YO(9,5)+48*YO(9,114)-18*YO(9,115)+18*YO(9,116)-164*YO(9,120), YO(9
,6)-6*YO(9,114)+6*YO(9,115)-9*YO(9,116)+83*YO(9,120), YO(9,7)-12*YO(9,114)+9*
YO(9,115)+6*YO(9,116)+81*YO(9,120), YO(9,8)+3*YO(9,114)-3*YO(9,115)-3*YO(9,116
)+YO(9,120), YO(9,9)-6*YO(9,114)-12*YO(9,115)-82*YO(9,120), YO(9,10)+9*YO(9,
114)-12*YO(9,116)-YO(9,120), YO(9,11)+39*YO(9,114)+9*YO(9,115)+27*YO(9,116)+9*
YO(9,118)+237*YO(9,120), YO(9,12)+15*YO(9,114)+18*YO(9,116)+9*YO(9,119)+651*YO
(9,120), YO(9,13)+39*YO(9,114)-24*YO(9,115)-6*YO(9,116)+9*YO(9,117)+9*YO(9,118
)+400*YO(9,120), YO(9,14)+27*YO(9,114)-27*YO(9,115)-30*YO(9,119)+244*YO(9,120)
, YO(9,15)+96*YO(9,114)-18*YO(9,115)-18*YO(9,116)-30*YO(9,117)-30*YO(9,118)+2*
YO(9,120), YO(9,16)+21*YO(9,114)+24*YO(9,115)+21*YO(9,116)-9*YO(9,118)+326*YO(
9,120), YO(9,17)-81*YO(9,114)-30*YO(9,115)-54*YO(9,116)+9*YO(9,117)+9*YO(9,118
)-82*YO(9,120), YO(9,18)+36*YO(9,114)+12*YO(9,116)+9*YO(9,118)+158*YO(9,120),
YO(9,19)-9*YO(9,114)+21*YO(9,115)+54*YO(9,116)-9*YO(9,117)-9*YO(9,118)+9*YO(9,
119)+337*YO(9,120), YO(9,20)-30*YO(9,114)+6*YO(9,115)+9*YO(9,116)+9*YO(9,119)+
233*YO(9,120), YO(9,21)+42*YO(9,114)-36*YO(9,115)-39*YO(9,116)-24*YO(9,117)-24
*YO(9,118)+239*YO(9,120), YO(9,22)-12*YO(9,114)+3*YO(9,115)-12*YO(9,119)-259*
YO(9,120), YO(9,23)-57*YO(9,114)+45*YO(9,115)+24*YO(9,116)-12*YO(9,118)+168*YO
(9,120), YO(9,24)+33*YO(9,114)+9*YO(9,115)-3*YO(9,116)-6*YO(9,118)+79*YO(9,120
), YO(9,25)-150*YO(9,114)+69*YO(9,115)-9*YO(9,116)-12*YO(9,118)+496*YO(9,120),
YO(9,26)+30*YO(9,114)+21*YO(9,115)+63*YO(9,116)-12*YO(9,117)-12*YO(9,118)-164*
YO(9,120), YO(9,27)+84*YO(9,114)+12*YO(9,115)+30*YO(9,116)-12*YO(9,119)-418*YO
(9,120), YO(9,28)+165*YO(9,114)-60*YO(9,115)+36*YO(9,116)+3*YO(9,117)+3*YO(9,
118)-814*YO(9,120), YO(9,29)+102*YO(9,114)-9*YO(9,115)+39*YO(9,116)-12*YO(9,
119)-414*YO(9,120), YO(9,30)-33*YO(9,114)-51*YO(9,115)-54*YO(9,116)+18*YO(9,
117)+27*YO(9,118)-9*YO(9,119)-346*YO(9,120), YO(9,31)-93*YO(9,114)+9*YO(9,115)
-27*YO(9,116)-9*YO(9,118)+18*YO(9,119)+655*YO(9,120), YO(9,32)-45*YO(9,114)+3*
YO(9,115)-42*YO(9,116)-21*YO(9,117)-30*YO(9,119)+165*YO(9,120), YO(9,33)+90*YO
(9,114)+51*YO(9,115)+57*YO(9,116)-3*YO(9,117)-33*YO(9,118)-27*YO(9,119)+425*YO
(9,120), YO(9,34)-9*YO(9,114)+6*YO(9,116)-161*YO(9,120), YO(9,35)+24*YO(9,114)
-12*YO(9,115)+18*YO(9,116)-18*YO(9,117)-18*YO(9,118)+168*YO(9,120), YO(9,36)+
105*YO(9,114)-57*YO(9,115)+33*YO(9,116)-15*YO(9,119)+269*YO(9,120), YO(9,37)-\
60*YO(9,114)+24*YO(9,115)-21*YO(9,116)-54*YO(9,119)+579*YO(9,120), YO(9,38)+3*
YO(9,114)-6*YO(9,115)+75*YO(9,116)+18*YO(9,117)+18*YO(9,118)-492*YO(9,120), YO
(9,39)+129*YO(9,114)+9*YO(9,115)+18*YO(9,116)+18*YO(9,117)+18*YO(9,118)+314*YO
(9,120), YO(9,40)-60*YO(9,114)-6*YO(9,115)-15*YO(9,116)+24*YO(9,117)+24*YO(9,
118)-325*YO(9,120), YO(9,41)-261*YO(9,114)-48*YO(9,115)-9*YO(9,116)+6*YO(9,117
)-12*YO(9,118)-694*YO(9,120), YO(9,42)-267*YO(9,114)+120*YO(9,115)-48*YO(9,116
)-6*YO(9,117)+33*YO(9,118)+3*YO(9,119)+743*YO(9,120), YO(9,43)+3*YO(9,114)-24*
YO(9,115)-42*YO(9,116)-21*YO(9,117)-24*YO(9,119)-84*YO(9,120), YO(9,44)-30*YO(
9,114)-3*YO(9,115)-27*YO(9,116)+6*YO(9,119)+82*YO(9,120), YO(9,45)+66*YO(9,114
)-3*YO(9,115)-3*YO(9,116)-18*YO(9,117)-18*YO(9,118)-580*YO(9,120), YO(9,46)+
183*YO(9,114)-27*YO(9,115)+96*YO(9,116)-12*YO(9,117)-39*YO(9,118)+42*YO(9,119)
+86*YO(9,120), YO(9,47)+57*YO(9,114)-36*YO(9,115)-21*YO(9,116)-45*YO(9,119)-\
650*YO(9,120), YO(9,48)-48*YO(9,114)-3*YO(9,116)+YO(9,120), YO(9,49)-150*YO(9,
114)+48*YO(9,115)-24*YO(9,116)+578*YO(9,120), YO(9,50)+102*YO(9,114)-54*YO(9,
115)+57*YO(9,116)-39*YO(9,119)-575*YO(9,120), YO(9,51)+21*YO(9,114)-27*YO(9,
115)-132*YO(9,116)-33*YO(9,117)-45*YO(9,118)+6*YO(9,119)+881*YO(9,120), YO(9,
52)-138*YO(9,114)+12*YO(9,115)-24*YO(9,116)+30*YO(9,117)+30*YO(9,118)+15*YO(9,
119)-568*YO(9,120), YO(9,53)+27*YO(9,114)-12*YO(9,115)+15*YO(9,116)+12*YO(9,
118)+33*YO(9,119)-16*YO(9,120), YO(9,54)-30*YO(9,114)-30*YO(9,115)-18*YO(9,116
)-12*YO(9,119)+334*YO(9,120), YO(9,55)+90*YO(9,114)+18*YO(9,115)+3*YO(9,116)+3
*YO(9,117)+21*YO(9,118)+18*YO(9,119)-106*YO(9,120), YO(9,56)+66*YO(9,114)-21*
YO(9,115)+12*YO(9,116)+42*YO(9,118)+30*YO(9,119)-1733*YO(9,120), YO(9,57)-276*
YO(9,114)-78*YO(9,115)+48*YO(9,116)+69*YO(9,117)+69*YO(9,118)+9*YO(9,119)+2071
*YO(9,120), YO(9,58)-210*YO(9,114)+66*YO(9,115)+33*YO(9,116)-24*YO(9,117)-12*
YO(9,118)-27*YO(9,119)-1367*YO(9,120), YO(9,59)-168*YO(9,114)-18*YO(9,115)-3*
YO(9,116)+15*YO(9,117)+6*YO(9,118)+99*YO(9,119)-1240*YO(9,120), YO(9,60)+15*YO
(9,114)-48*YO(9,115)-30*YO(9,116)+15*YO(9,117)-15*YO(9,118)+30*YO(9,119)+334*
YO(9,120), YO(9,61)-177*YO(9,114)+54*YO(9,115)-54*YO(9,116)+33*YO(9,117)+24*YO
(9,118)-21*YO(9,119)-968*YO(9,120), YO(9,62)+30*YO(9,114)+18*YO(9,115)-33*YO(9
,116)-18*YO(9,117)-6*YO(9,118)+6*YO(9,119)-167*YO(9,120), YO(9,63)-99*YO(9,114
)+30*YO(9,115)+15*YO(9,116)-24*YO(9,117)-24*YO(9,118)-140*YO(9,120), YO(9,64)-\
66*YO(9,114)+45*YO(9,115)-33*YO(9,116)+6*YO(9,117)+3*YO(9,118)+18*YO(9,119)+
496*YO(9,120), YO(9,65)-87*YO(9,114)-6*YO(9,115)+24*YO(9,116)-715*YO(9,120), 
YO(9,66)-120*YO(9,114)-66*YO(9,115)+78*YO(9,116)-9*YO(9,117)+12*YO(9,118)+111*
YO(9,119)+1577*YO(9,120), YO(9,67)+102*YO(9,114)+9*YO(9,115)+9*YO(9,116)-18*YO
(9,117)-9*YO(9,118)+39*YO(9,119)-578*YO(9,120), YO(9,68)-222*YO(9,114)+21*YO(9
,115)-15*YO(9,116)+21*YO(9,117)-6*YO(9,118)+60*YO(9,119)-95*YO(9,120), YO(9,69
)-273*YO(9,114)+78*YO(9,115)-90*YO(9,116)+21*YO(9,117)+15*YO(9,118)-36*YO(9,
119)-1134*YO(9,120), YO(9,70)+408*YO(9,114)+3*YO(9,115)+66*YO(9,116)-3*YO(9,
117)-102*YO(9,118)+72*YO(9,119)-282*YO(9,120), YO(9,71)+168*YO(9,114)-3*YO(9,
115)+96*YO(9,116)+15*YO(9,117)-24*YO(9,118)+27*YO(9,119)-340*YO(9,120), YO(9,
72)+309*YO(9,114)+39*YO(9,115)+108*YO(9,116)+36*YO(9,118)+66*YO(9,119)-2244*YO
(9,120), YO(9,73)+186*YO(9,114)+39*YO(9,116)+12*YO(9,117)+12*YO(9,118)+225*YO(
9,120), YO(9,74)-48*YO(9,114)+39*YO(9,115)+15*YO(9,116)+51*YO(9,119)+156*YO(9,
120), YO(9,75)+15*YO(9,114)+36*YO(9,115)-24*YO(9,116)-9*YO(9,117)-66*YO(9,118)
+66*YO(9,119)+2*YO(9,120), YO(9,76)+6*YO(9,114)-18*YO(9,115)+66*YO(9,116)+24*
YO(9,117)-15*YO(9,118)-15*YO(9,119)+1000*YO(9,120), YO(9,77)-180*YO(9,114)-48*
YO(9,115)-126*YO(9,116)-15*YO(9,117)+81*YO(9,118)-87*YO(9,119)+99*YO(9,120), 
YO(9,78)+381*YO(9,114)-63*YO(9,115)+57*YO(9,116)-27*YO(9,117)-48*YO(9,118)+9*
YO(9,119)-110*YO(9,120), YO(9,79)-204*YO(9,114)+30*YO(9,115)-120*YO(9,116)-6*
YO(9,117)-21*YO(9,118)-18*YO(9,119)-142*YO(9,120), YO(9,80)-195*YO(9,114)+63*
YO(9,115)-60*YO(9,116)+12*YO(9,117)-9*YO(9,118)+24*YO(9,119)-142*YO(9,120), YO
(9,81)+255*YO(9,114)-57*YO(9,115)+90*YO(9,116)+12*YO(9,117)-30*YO(9,118)+171*
YO(9,119)-412*YO(9,120), YO(9,82)-156*YO(9,114)-30*YO(9,115)+36*YO(9,117)+36*
YO(9,119)-154*YO(9,120), YO(9,83)+18*YO(9,114)-15*YO(9,115)+15*YO(9,116)+24*YO
(9,117)-6*YO(9,118)-33*YO(9,119)-812*YO(9,120), YO(9,84)+183*YO(9,114)+30*YO(9
,115)+12*YO(9,116)-6*YO(9,117)+78*YO(9,118)-123*YO(9,119)-595*YO(9,120), YO(9,
85)-78*YO(9,114)+12*YO(9,115)-36*YO(9,116)-48*YO(9,117)-21*YO(9,118)+12*YO(9,
119)+1808*YO(9,120), YO(9,86)+132*YO(9,114)-15*YO(9,115)+45*YO(9,116)-24*YO(9,
119)-892*YO(9,120), YO(9,87)-258*YO(9,114)-3*YO(9,115)-45*YO(9,116)+72*YO(9,
117)+108*YO(9,118)-93*YO(9,119)-592*YO(9,120), YO(9,88)-81*YO(9,114)+18*YO(9,
115)+57*YO(9,116)+45*YO(9,117)+54*YO(9,118)-54*YO(9,119)-741*YO(9,120), YO(9,
89)-99*YO(9,114)+24*YO(9,115)-96*YO(9,116)+48*YO(9,117)-12*YO(9,118)+69*YO(9,
119)-2205*YO(9,120), YO(9,90)+33*YO(9,114)-114*YO(9,115)+201*YO(9,116)+39*YO(9
,117)+18*YO(9,118)+6*YO(9,119)+991*YO(9,120), YO(9,91)-144*YO(9,114)+60*YO(9,
115)-6*YO(9,116)+414*YO(9,120), YO(9,92)-75*YO(9,114)+54*YO(9,115)-30*YO(9,118
)-24*YO(9,119)+1157*YO(9,120), YO(9,93)-90*YO(9,114)-27*YO(9,115)+51*YO(9,116)
+45*YO(9,118)+6*YO(9,119)+342*YO(9,120), YO(9,94)-234*YO(9,114)+24*YO(9,115)-\
39*YO(9,116)+9*YO(9,117)+42*YO(9,118)-72*YO(9,119)-330*YO(9,120), YO(9,95)-327
*YO(9,114)+96*YO(9,115)+3*YO(9,116)+30*YO(9,117)+18*YO(9,118)-6*YO(9,119)+2224
*YO(9,120), YO(9,96)-81*YO(9,114)-54*YO(9,115)-21*YO(9,116)-3*YO(9,117)+39*YO(
9,118)+84*YO(9,119)+147*YO(9,120), YO(9,97)+6*YO(9,114)-3*YO(9,115)+18*YO(9,
116)+24*YO(9,118)-245*YO(9,120), YO(9,98)-348*YO(9,114)-45*YO(9,115)-129*YO(9,
116)+60*YO(9,117)+57*YO(9,118)+1394*YO(9,120), YO(9,99)-6*YO(9,114)-51*YO(9,
115)-69*YO(9,116)+42*YO(9,117)-21*YO(9,118)+6*YO(9,119)-856*YO(9,120), YO(9,
100)-342*YO(9,114)-108*YO(9,115)-102*YO(9,116)+72*YO(9,117)+117*YO(9,118)+51*
YO(9,119)-1082*YO(9,120), YO(9,101)+183*YO(9,114)+21*YO(9,115)-15*YO(9,116)-69
*YO(9,117)-24*YO(9,118)-51*YO(9,119)+322*YO(9,120), YO(9,102)+612*YO(9,114)-9*
YO(9,115)-78*YO(9,116)-60*YO(9,117)-255*YO(9,118)+126*YO(9,119)-499*YO(9,120),
YO(9,103)+45*YO(9,114)-12*YO(9,115)-18*YO(9,119)-741*YO(9,120), YO(9,104)+411*
YO(9,114)+63*YO(9,115)+24*YO(9,116)-78*YO(9,118)+102*YO(9,119)+870*YO(9,120),
YO(9,105)-168*YO(9,114)+87*YO(9,115)-6*YO(9,116)+57*YO(9,117)+156*YO(9,118)-\
165*YO(9,119)-266*YO(9,120), YO(9,106)-537*YO(9,114)-18*YO(9,115)+27*YO(9,116)
+51*YO(9,117)+216*YO(9,118)-138*YO(9,119)-1733*YO(9,120), YO(9,107)+387*YO(9,
114)-45*YO(9,115)+144*YO(9,116)-36*YO(9,117)-168*YO(9,118)-45*YO(9,119)+3298*
YO(9,120), YO(9,108)+483*YO(9,114)-84*YO(9,115)+96*YO(9,116)+45*YO(9,117)-135*
YO(9,118)+39*YO(9,119)+58*YO(9,120), YO(9,109)-357*YO(9,114)-33*YO(9,115)-180*
YO(9,116)-15*YO(9,117)+30*YO(9,118)+72*YO(9,119)+2073*YO(9,120), YO(9,110)-330
*YO(9,114)+99*YO(9,115)-186*YO(9,116)+138*YO(9,117)+159*YO(9,118)+45*YO(9,119)
-2747*YO(9,120), YO(9,111)+312*YO(9,114)+87*YO(9,115)+87*YO(9,116)-27*YO(9,117
)+45*YO(9,118)-1923*YO(9,120), YO(9,112)+237*YO(9,114)-183*YO(9,115)-9*YO(9,
116)+27*YO(9,117)+207*YO(9,118)-87*YO(9,119)+1620*YO(9,120), YO(9,113)-396*YO(
9,114)-186*YO(9,115)-108*YO(9,116)+45*YO(9,117)-30*YO(9,118)+252*YO(9,119)+
1433*YO(9,120)]:
#
# The coefficients are always a multiple of 3, except for the
# YO(9,120) [the z6 guy], which is an integer, but can be (say) 1.
#