#######################################################################
# 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 provides tables of coproducts of the L-loop MHV amplitude at
# (1,1,1), particularly those above weight 11, because they are the
# boundary conditions for integrating up the amplitude everywhere else.
# Here {\cal E}^{(L)} = EZMHVgL is the L-loop MHV amplitude,
# and we use the {\cal L}^a_hex letters
#
#    {  a,     b,     c,     mu,    mv,    mw,  yu, yv, yw } 
# = { a-hat, b-hat, c-hat, d-hat, e-hat, f-hat, yu, yv, yw } in the paper. 
#
# Here
# "zi" = zeta[i], the Riemann zeta value.
# Zeta53 = zeta[5,3], the multiple zeta value.
# Zeta113 = zeta[11,3] and Zeta115 = zeta[11,5], in a real abuse of notation.
#
# We will give all the results a second time in the f-alphabet
# (see the ancillary file "ftoMZV16.txt" for the conversions).
#
#########################################
# We use the new, all orders cosmic normalization factor "rho",
# which through 8 loops evaluates to:
#
rho := 1+(-z4)*gsq^2+(50/3*z6+8*z3^2)*gsq^3+(-2891/12*z8-160*z3*z5)*gsq^4
+(10265/3*z10-40*z4*z3^2+912*z5^2+1680*z3*z7)*gsq^5
+(-4857061891/99504*z12+1600/3*z6*z3^2+608*z4*z3*z5-20832*z5*z7-18816*z3*z9)
  *gsq^6
+(126240*z7^2+50570065/72*z14-6370*z8*z3^2-2736*z4*z5^2+247296*z5*z9
 +221760*z3*z11-24320/3*z6*z3*z5-5040*z4*z3*z7)*gsq^7
+(34880*z6*z5^2+320*z4*z3^4-3130560*z7*z9-3041280*z5*z11-2718144*z3*z13
 -63718004141707/6250176*z16+230960/3*z10*z3^2+98056*z8*z3*z5
 +67840*z6*z3*z7+52128*z4*z5*z7+45696*z4*z3*z9)*gsq^8 :
#
# where gsq = g^2. 
#
# For reference, this was the old version of rho that we constructed
# using the coaction principle through 7 loops:
#
rho_OLD := 1 + 8 * z3^2 * gsq^3 - 160 * z3 * z5 * gsq^4
+ ( 1680 * z3 * z7 + 912 * z5^2 - 32 * z4 * z3^2 ) * gsq^5
- ( 18816 * z3 * z9 + 20832 * z5 * z7
  - 448 * z4 * z3 * z5 - 400 * z6 * z3^2 ) * gsq^6
+ ( 221760 * z3 * z11 + 247296 * z5 * z9 + 126240 * z7^2
   - 3360 * z4 * z3 * z7 - 1824 * z4 * z5^2 - 5440 * z6 * z3 * z5
   - 4480 * z8 * z3^2 ) * gsq^7 :
#
# It was used to normalize the amplitudes in
# arXiv:1903.10890 and arXiv:1906.07116.
# To go from that normalization to the current one, multiply by
#
# rho_OLD/rho = XtoZgfactor,
#
XtoZgfactor := 1+z4*gsq^2+(-50/3*z6)*gsq^3+2905/12*z8*gsq^4
+(-10375/3*z10)*gsq^5+4937878055/99504*z12*gsq^6
+(-51698725/72*z14)*gsq^7 :
#
###########################################################################
###########################################################################
# RESULTS IN TERMS OF "CONVENTIONAL" MZVs #
###########################################
# First we give the value of the full MHV amplitude at (1,1,1)
# through 8 loops.
#
EZMHVg0_111 := 1 :
EZMHVg1_111 := 0 :
EZMHVg2_111 := - 9 * z4 :
EZMHVg3_111 :=  121 * z6 :
EZMHVg4_111 := - 6381/4 * z8 + 24 * ( Zeta53 + 5 * z3 * z5 - z2 * z3^2 ) :
#
EZMHVg5_111 := 222069/10 * z10 - 12 * ( 4 * z2 * Zeta53 + 25 * z5^2 )
- 96 * ( 2 * Zeta73 + 28 * z3 * z7 + 11 * z5^2 - 4 * z2 * z3 * z5
       - 6 * z4 * z3^2 ) :
#
EZMHVg6_111 := - 10709056285/33168 * z12
  + 16176 * ( 1/9 * Zeta93 + 3 * z3 * z9 - 40/3 * z5 * z7
            + 6 * z2 * z3 * z7 + z2 * z5^2 + 6 * z4 * z3 * z5 - 6 * z6 * z3^2 )
  + 384 * z2 * ( Zeta73 + 14 * z3 * z7 + 3 * z5^2 - 4 * z4 * z3^2 )
    + 504 * z4 * ( Zeta53 + 5 * z3 * z5 - z2 * z3^2 )
  + 1884 * ( 140 * z5 * z7 - 56 * z2 * z3 * z7 - 10 * z2 * z5^2
           - 60 * z4 * z3 * z5 + 49 * z6 * z3^2 ) :
#
EZMHVg7_111 := 7113137749/5040 * z14
  + 2496 * ( 2 * Zeta533 * z3 - 2 * Zeta5333 - Zeta53 * z3^2 )
  - 87648 * Zeta95 + 302160 * Zeta113 - 61024/3 * z2 * Zeta93
    - 3456 * z4 * Zeta73 - 7644 * z6 * Zeta53 
  + 121912 * z8 * z3^2 + 416 * z2 * z3^4 + 204888 * z6 * z3 * z5
    - 4160 * z3^3 * z5
  + 93780 * z4 * z5^2 + 200448 * z4 * z3 * z7
    - 77408 * z2 * z5 * z7 + 1208712 * z7^2
  + 252640 * z2 * z3 * z9 + 1082048 * z5 * z9 - 1241760 * z3 * z11 :
#
EZMHVg8_111 := - 512193667550809/7639104 * z16
+ 576 * Zeta53^2 - 2764512 * Zeta133 + 4901904/11 * Zeta115
+ 58944 * Zeta7333 - 54720 * Zeta5533 + 4224 * Zeta53 * z2 * z3^2
- 8448 * z3 * z2 * Zeta533 + 6144 * z2 * Zeta5333 - 19648 * z3^3 * z2 * z5
+ 5485728 * z7^2 * z2 + 1799904 * z2 * Zeta113 - 411968 * z2 * Zeta95
+ 259312 * z4 * Zeta93 + 5760 * z5 * Zeta533 + 56784 * z6 * Zeta73
+ 57930 * z8 * Zeta53 + 137536 * z3^3 * z7 + 188496 * z3^2 * z5^2
+ 29472 * z3^2 * Zeta73 + 54720 * z3 * Zeta553 - 58944 * z3 * Zeta733
- 19560489/10 * z10 * z3^2 - 1550370 * z6 * z5^2 - 14736 * z4 * z3^4
+ 23915376 * z3 * z13 + 3016464 * z5 * z11 - 57222368/11 * z7 * z9
- 4161408 * z3 * z11 * z2 + 7035328 * z5 * z9 * z2 - 4448304 * z4 * z3 * z9
- 3421414 * z8 * z3 * z5 - 3217728 * z6 * z3 * z7 - 1910928 * z4 * z5 * z7 :
#
################################################################
# We also specify the independent, nonvanishing multiple coproducts
# at (1,1,1), when they have weight above 11 so they cannot be
# read out of the YE tables.
#
#################################################################
# SINGLES #
###########
# Because of a final-entry conditions, branch-cut conditions, and parity,
# the MHV single coproducts all vanish at (1,1,1).
#
# The 6 independent weight 13 single final entries at 7 loops:
#
EZMHVg7c_111[mu] := 0: 
EZMHVg7c_111[mv] := 0: 
EZMHVg7c_111[mw] := 0: 
EZMHVg7c_111[yu] := 0: 
EZMHVg7c_111[yv] := 0: 
EZMHVg7c_111[yw] := 0: 
#
# The 6 independent weight 15 single final entries at 8 loops:
#
EZMHVg8c_111[mu] := 0: 
EZMHVg8c_111[mv] := 0: 
EZMHVg8c_111[mw] := 0: 
EZMHVg8c_111[yu] := 0: 
EZMHVg8c_111[yv] := 0: 
EZMHVg8c_111[yw] := 0: 
#
#################################################################
# DOUBLES #
###########
# Next we provide the double coproducts, first at 7 loops,
# where they are weight 14 - 2 = 12.
#
# The 21 linearly independent weight 12 double coproducts at (1,1,1)
# at 7 loops:
#
EZMHVg7dc_111[a,mv] := -9048*z4*z3*z5-1920*z2*z3*z7-4758*z6*z3^2+29616*z3*z9
-944*z2*z5^2+85600/3*z5*z7+208*z2*Zeta73+72*z4*Zeta53-41184375511/199008*z12
+9872/9*Zeta93:
EZMHVg7dc_111[b,mu] := -9048*z4*z3*z5-1920*z2*z3*z7-4758*z6*z3^2+29616*z3*z9
-944*z2*z5^2+85600/3*z5*z7+208*z2*Zeta73+72*z4*Zeta53-41184375511/199008*z12
+9872/9*Zeta93:
EZMHVg7dc_111[c,mu] := -9048*z4*z3*z5-1920*z2*z3*z7-4758*z6*z3^2+29616*z3*z9
-944*z2*z5^2+85600/3*z5*z7+208*z2*Zeta73+72*z4*Zeta53-41184375511/199008*z12
+9872/9*Zeta93:
EZMHVg7dc_111[c,yv] := 0:
EZMHVg7dc_111[mu,mu] := 0:
EZMHVg7dc_111[mu,mv] := 0:
EZMHVg7dc_111[mv,mv] := 0:
EZMHVg7dc_111[mv,mw] := 0:
EZMHVg7dc_111[mv,yw] := 0:
EZMHVg7dc_111[mw,mu] := 0:
EZMHVg7dc_111[mw,mw] := 0:
EZMHVg7dc_111[mw,yw] := 0:
EZMHVg7dc_111[yu,mw] := 0:
EZMHVg7dc_111[yv,mv] := 0:
EZMHVg7dc_111[yv,mw] := 0:
EZMHVg7dc_111[yw,mu] := 0:
EZMHVg7dc_111[yw,mv] := 0:
EZMHVg7dc_111[yw,mw] := 0:
EZMHVg7dc_111[yw,yu] := 27366*z6*z3^2+4808*z2*z5^2-3384*z4*Zeta53
-2176*z2*Zeta73+10560*z2*z3*z7+51096*z4*z3*z5-588400/3*z5*z7-168432*z3*z9
+85695597061/49752*z12-56144/9*Zeta93:
EZMHVg7dc_111[yw,yv] := 27366*z6*z3^2+4808*z2*z5^2-3384*z4*Zeta53
-2176*z2*Zeta73+10560*z2*z3*z7+51096*z4*z3*z5-588400/3*z5*z7-168432*z3*z9
+85695597061/49752*z12-56144/9*Zeta93:
EZMHVg7dc_111[yw,yw] := 27366*z6*z3^2+4808*z2*z5^2-3384*z4*Zeta53
-2176*z2*Zeta73+10560*z2*z3*z7+51096*z4*z3*z5-588400/3*z5*z7-168432*z3*z9
+85695597061/49752*z12-56144/9*Zeta93:
#
# The 21 linearly independent weight 14 double coproducts at (1,1,1)
# at 8 loops:
#
EZMHVg8dc_111[a,mv] := -5632*Zeta5333+346768*Zeta113-98768*Zeta95
-42437345879/60480*z14+171036*z6*z3*z5+197232*z4*z3*z7+1488352*z7^2
-197344/9*z2*Zeta93-1080*z4*Zeta73-2816*Zeta53*z3^2+5632*Zeta533*z3
+1408/3*z2*z3^4-2292*z6*Zeta53+94344*z4*z5^2-14080/3*z3^3*z5+258884/3*z8*z3^2
+4434256/3*z5*z9-1123408*z3*z11+836000/3*z2*z3*z9-284384/3*z2*z5*z7:
EZMHVg8dc_111[b,mu] := -5632*Zeta5333+346768*Zeta113-98768*Zeta95
-42437345879/60480*z14+171036*z6*z3*z5+197232*z4*z3*z7+1488352*z7^2
-197344/9*z2*Zeta93-1080*z4*Zeta73-2816*Zeta53*z3^2+5632*Zeta533*z3
+1408/3*z2*z3^4-2292*z6*Zeta53+94344*z4*z5^2-14080/3*z3^3*z5+258884/3*z8*z3^2
+4434256/3*z5*z9-1123408*z3*z11+836000/3*z2*z3*z9-284384/3*z2*z5*z7:
EZMHVg8dc_111[c,mu] := -5632*Zeta5333+346768*Zeta113-98768*Zeta95
-42437345879/60480*z14+171036*z6*z3*z5+197232*z4*z3*z7+1488352*z7^2
-197344/9*z2*Zeta93-1080*z4*Zeta73-2816*Zeta53*z3^2+5632*Zeta533*z3
+1408/3*z2*z3^4-2292*z6*Zeta53+94344*z4*z5^2-14080/3*z3^3*z5+258884/3*z8*z3^2
+4434256/3*z5*z9-1123408*z3*z11+836000/3*z2*z3*z9-284384/3*z2*z5*z7:
EZMHVg8dc_111[c,yv] := 0: EZMHVg8dc_111[mu,mu] := 0:
EZMHVg8dc_111[mu,mv] := 0: EZMHVg8dc_111[mv,mv] := 0:
EZMHVg8dc_111[mv,mw] := 0: EZMHVg8dc_111[mv,yw] := 0:
EZMHVg8dc_111[mw,mu] := 0: EZMHVg8dc_111[mw,mw] := 0:
EZMHVg8dc_111[mw,yw] := 0: EZMHVg8dc_111[yu,mw] := 0:
EZMHVg8dc_111[yv,mv] := 0: EZMHVg8dc_111[yv,mw] := 0:
EZMHVg8dc_111[yw,mu] := 0: EZMHVg8dc_111[yw,mv] := 0:
EZMHVg8dc_111[yw,mw] := 0:
EZMHVg8dc_111[yw,yu] := -3675296/3*z2*z3*z9-932664*z6*z3*z5+1256864/3*z2*z5*z7
-1028544*z4*z3*z7-24448*Zeta533*z3+57204*z6*Zeta53+12224*Zeta53*z3^2
-1389512/3*z8*z3^2+981664/9*z2*Zeta93-15172576/3*z5*z9+5593216*z3*z11
-518604*z4*z5^2+61120/3*z3^3*z5+35424*z4*Zeta73-6112/3*z2*z3^4
+24448*Zeta5333-1496656*Zeta113+430784*Zeta95-323971645187/30240*z14
-5677624*z7^2:
EZMHVg8dc_111[yw,yv] := -3675296/3*z2*z3*z9-932664*z6*z3*z5+1256864/3*z2*z5*z7
-1028544*z4*z3*z7-24448*Zeta533*z3+57204*z6*Zeta53+12224*Zeta53*z3^2
-1389512/3*z8*z3^2+981664/9*z2*Zeta93-15172576/3*z5*z9+5593216*z3*z11
-518604*z4*z5^2+61120/3*z3^3*z5+35424*z4*Zeta73-6112/3*z2*z3^4
+24448*Zeta5333-1496656*Zeta113+430784*Zeta95-323971645187/30240*z14
-5677624*z7^2:
EZMHVg8dc_111[yw,yw] := -3675296/3*z2*z3*z9-932664*z6*z3*z5+1256864/3*z2*z5*z7
-1028544*z4*z3*z7-24448*Zeta533*z3+57204*z6*Zeta53+12224*Zeta53*z3^2
-1389512/3*z8*z3^2+981664/9*z2*Zeta93-15172576/3*z5*z9+5593216*z3*z11
-518604*z4*z5^2+61120/3*z3^3*z5+35424*z4*Zeta73-6112/3*z2*z3^4
+24448*Zeta5333-1496656*Zeta113+430784*Zeta95-323971645187/30240*z14
-5677624*z7^2:
#
# The remaining doubles can be found using "Edouble", or the relations
# "doublerels33" in "MHV8quintuples.txt".
#
#################################################################
# TRIPLES #
###########
# Next we provide the triple coproducts at (1,1,1).  Since at 7 loops
# they are weight 11, we just have to do this at 8 loops:
#
# The 62 linearly independent weight 13 triple coproducts at (1,1,1)
# at 8 loops:
#
EZMHVg8tc_111[a,a,mv] := 2456*z2*z3^2*z5-352*Zeta53*z2*z3+173392*z11*z2
+185346*z4*z9+352*z2*Zeta533+2456*z4*z3^3+6520163/40*z10*z3-2456*z3*Zeta73
+134072*z6*z7+1710707/12*z8*z5-17192*z3^2*z7-15708*z3*z5^2+2456*Zeta733
-2280*Zeta553-996474*z13:
EZMHVg8tc_111[a,mv,yw] := 0: EZMHVg8tc_111[a,mw,mw] := 0: 
EZMHVg8tc_111[a,yv,mv] := 0: EZMHVg8tc_111[a,yv,mw] := 0:
EZMHVg8tc_111[b,b,mu] := 2456*z2*z3^2*z5-352*Zeta53*z2*z3+173392*z11*z2
+185346*z4*z9+352*z2*Zeta533+2456*z4*z3^3+6520163/40*z10*z3-2456*z3*Zeta73
+134072*z6*z7+1710707/12*z8*z5-17192*z3^2*z7-15708*z3*z5^2+2456*Zeta733
-2280*Zeta553-996474*z13: 
EZMHVg8tc_111[b,mu,mu] := 0: EZMHVg8tc_111[b,mw,yw] := 0:
EZMHVg8tc_111[c,c,mu] := 2456*z2*z3^2*z5-352*Zeta53*z2*z3+173392*z11*z2
+185346*z4*z9+352*z2*Zeta533+2456*z4*z3^3+6520163/40*z10*z3-2456*z3*Zeta73
+134072*z6*z7+1710707/12*z8*z5-17192*z3^2*z7-15708*z3*z5^2+2456*Zeta733
-2280*Zeta553-996474*z13: 
EZMHVg8tc_111[c,c,yv] := 0: EZMHVg8tc_111[c,mu,mv] := 0:
EZMHVg8tc_111[c,yv,mv] := 0: EZMHVg8tc_111[mu,b,mu] := 0:
EZMHVg8tc_111[mu,c,mu] := 0: EZMHVg8tc_111[mu,c,yv] := 0:
EZMHVg8tc_111[mu,mu,mu] := 0: EZMHVg8tc_111[mu,mu,mv] := 0: 
EZMHVg8tc_111[mu,mv,mv] := 0: EZMHVg8tc_111[mu,mw,mu] := 0:
EZMHVg8tc_111[mu,mw,yw] := 0: EZMHVg8tc_111[mu,yv,mw] := 0:
EZMHVg8tc_111[mu,yw,mu] := 0: EZMHVg8tc_111[mu,yw,mv] := 0:
EZMHVg8tc_111[mu,yw,mw] := 0: EZMHVg8tc_111[mv,a,mv] := 0:
EZMHVg8tc_111[mv,c,mu] := 0: EZMHVg8tc_111[mv,c,yv] := 0: 
EZMHVg8tc_111[mv,mu,mv] := 0: EZMHVg8tc_111[mv,mv,mv] := 0:
EZMHVg8tc_111[mv,mv,mw] := 0: EZMHVg8tc_111[mv,mv,yw] := 0:
EZMHVg8tc_111[mv,mw,mu] := 0: EZMHVg8tc_111[mv,yu,mw] := 0:
EZMHVg8tc_111[mv,yv,mv] := 0: EZMHVg8tc_111[mv,yw,mu] := 0:
EZMHVg8tc_111[mv,yw,mv] := 0: EZMHVg8tc_111[mv,yw,mw] := 0: 
EZMHVg8tc_111[mw,a,mv] := 0: EZMHVg8tc_111[mw,b,mu] := 0:
EZMHVg8tc_111[mw,mw,mw] := 0: EZMHVg8tc_111[mw,yv,mv] := 0:
EZMHVg8tc_111[mw,yw,mw] := 0: EZMHVg8tc_111[yu,b,mu] := 0:
EZMHVg8tc_111[yu,mu,mu] := 0: EZMHVg8tc_111[yu,mu,mv] := 0:
EZMHVg8tc_111[yv,c,yv] := 2456*z2*z3^2*z5-352*Zeta53*z2*z3+173392*z11*z2
+185346*z4*z9+352*z2*Zeta533+2456*z4*z3^3+6520163/40*z10*z3-2456*z3*Zeta73
+134072*z6*z7+1710707/12*z8*z5-17192*z3^2*z7-15708*z3*z5^2+2456*Zeta733
-2280*Zeta553-996474*z13:
EZMHVg8tc_111[yv,mv,yw] := 0: EZMHVg8tc_111[yv,mw,yw] := 0:
EZMHVg8tc_111[yv,yu,mw] := 4912*z2*z3^2*z5-704*Zeta53*z2*z3+346784*z11*z2
+370692*z4*z9+704*z2*Zeta533+4912*z4*z3^3+6520163/20*z10*z3-4912*z3*Zeta73
+268144*z6*z7+1710707/6*z8*z5-34384*z3^2*z7-31416*z3*z5^2+4912*Zeta733
-4560*Zeta553-1992948*z13:
EZMHVg8tc_111[yv,yv,mv] := 4912*z2*z3^2*z5-704*Zeta53*z2*z3+346784*z11*z2
+370692*z4*z9+704*z2*Zeta533+4912*z4*z3^3+6520163/20*z10*z3-4912*z3*Zeta73
+268144*z6*z7+1710707/6*z8*z5-34384*z3^2*z7-31416*z3*z5^2+4912*Zeta733
-4560*Zeta553-1992948*z13:
EZMHVg8tc_111[yw,mu,mu] := 0: EZMHVg8tc_111[yw,mv,mw] := 0:
EZMHVg8tc_111[yw,mv,yw] := 0: EZMHVg8tc_111[yw,mw,mu] := 0: 
EZMHVg8tc_111[yw,mw,mw] := 0:
EZMHVg8tc_111[yw,yv,mv] := 4912*z2*z3^2*z5-704*Zeta53*z2*z3+346784*z11*z2
+370692*z4*z9+704*z2*Zeta533+4912*z4*z3^3+6520163/20*z10*z3-4912*z3*Zeta73
+268144*z6*z7+1710707/6*z8*z5-34384*z3^2*z7-31416*z3*z5^2+4912*Zeta733
-4560*Zeta553-1992948*z13:
EZMHVg8tc_111[yw,yv,mw] := 4912*z2*z3^2*z5-704*Zeta53*z2*z3+346784*z11*z2
+370692*z4*z9+704*z2*Zeta533+4912*z4*z3^3+6520163/20*z10*z3-4912*z3*Zeta73
+268144*z6*z7+1710707/6*z8*z5-34384*z3^2*z7-31416*z3*z5^2+4912*Zeta733
-4560*Zeta553-1992948*z13:
EZMHVg8tc_111[yw,yw,mu] := 4912*z2*z3^2*z5-704*Zeta53*z2*z3+346784*z11*z2
+370692*z4*z9+704*z2*Zeta533+4912*z4*z3^3+6520163/20*z10*z3-4912*z3*Zeta73
+268144*z6*z7+1710707/6*z8*z5-34384*z3^2*z7-31416*z3*z5^2+4912*Zeta733
-4560*Zeta553-1992948*z13:
EZMHVg8tc_111[yw,yw,mw] := 4912*z2*z3^2*z5-704*Zeta53*z2*z3+346784*z11*z2
+370692*z4*z9+704*z2*Zeta533+4912*z4*z3^3+6520163/20*z10*z3-4912*z3*Zeta73
+268144*z6*z7+1710707/6*z8*z5-34384*z3^2*z7-31416*z3*z5^2+4912*Zeta733
-4560*Zeta553-1992948*z13: 
EZMHVg8tc_111[yw,yw,yu] := 0: EZMHVg8tc_111[yw,yw,yv] := 0:
EZMHVg8tc_111[yw,yw,yw] := 0:
#
# The remaining triples can be found using "Etriple", or the relations
# "triplerels127" in "MHV8quintuples.txt".
#
#################################################################
# QUADRUPLES #
##############
# Next we provide the quadruple coproducts at (1,1,1).  Since at 7 loops
# they are weight 10, we just have to do this at 8 loops:
#
# The 166 linearly independent weight 12 quadruple coproducts at (1,1,1)
# at 8 loops:
#
EZMHVg8qc_111[a,a,a,mv] := 30812661671/16584*z12+104615/2*z6*z3^2+99210
*z4*z3*z5-1566*z4*Zeta53+21364*z2*z3*z7+10316*z2*z5^2-2508*z2*Zeta73-325710*z3*
z9-322140*z5*z7-36190/3*Zeta93: EZMHVg8qc_111[a,a,mv,yw] := 0: EZMHVg8qc_111[a
,a,mw,mw] := 160553227/2764*z12+1494*z6*z3^2+3600*z4*z3*z5-168*z4*Zeta53+648*z2*
z3*z7+284*z2*z5^2-152*z2*Zeta73-10260*z3*z9-12640*z5*z7-380*Zeta93: 
EZMHVg8qc_111[a,a,yv,mv] := 0: EZMHVg8qc_111[a,a,yv,mw] := 0: EZMHVg8qc_111[a,
mv,a,mv] := -4600*z2*z3*z7-2252*z2*z5^2-11248*z6*z3^2-20424*z4*z3*z5+144*z4*
Zeta53+424*z2*Zeta73+68988*z3*z9+194000/3*z5*z7+22996/9*Zeta93-75111271853/
199008*z12: EZMHVg8qc_111[a,mv,c,mu] := -5248*z2*z3*z7-2536*z2*z5^2-12742*z6*z3
^2-24024*z4*z3*z5+312*z4*Zeta53+576*z2*Zeta73+79248*z3*z9+231920/3*z5*z7+26416/
9*Zeta93-86671104197/199008*z12: EZMHVg8qc_111[a,mv,c,yv] := 0: EZMHVg8qc_111[
a,mv,mu,mv] := 2624*z2*z3*z7+1268*z2*z5^2+6371*z6*z3^2+12012*z4*z3*z5-156*z4*
Zeta53-288*z2*Zeta73-39624*z3*z9-115960/3*z5*z7-13208/9*Zeta93+86671104197/
398016*z12: EZMHVg8qc_111[a,mv,mv,mv] := -12363*z6*z3^2-2260*z2*z5^2+1500*z4*
Zeta53+1312*z2*Zeta73-29676*z4*z3*z5-5088*z2*z3*z7+105480*z5*z7+84312*z3*z9
-4134176867/132672*z12+9368/3*Zeta93: EZMHVg8qc_111[a,mv,mv,mw] := 0: 
EZMHVg8qc_111[a,mv,mv,yw] := 0: EZMHVg8qc_111[a,mv,yu,mw] := 0: EZMHVg8qc_111[
a,mw,a,mv] := -4600*z2*z3*z7-2252*z2*z5^2-11248*z6*z3^2-20424*z4*z3*z5+144*z4*
Zeta53+424*z2*Zeta73+68988*z3*z9+194000/3*z5*z7+22996/9*Zeta93-75111271853/
199008*z12: EZMHVg8qc_111[b,b,b,mu] := 30812661671/16584*z12+104615/2*z6*z3^2+
99210*z4*z3*z5-1566*z4*Zeta53+21364*z2*z3*z7+10316*z2*z5^2-2508*z2*Zeta73
-325710*z3*z9-322140*z5*z7-36190/3*Zeta93: EZMHVg8qc_111[b,b,mu,mu] := 160553227
/2764*z12+1494*z6*z3^2+3600*z4*z3*z5-168*z4*Zeta53+648*z2*z3*z7+284*z2*z5^2-152
*z2*Zeta73-10260*z3*z9-12640*z5*z7-380*Zeta93: EZMHVg8qc_111[b,b,mw,yw] := 0: 
EZMHVg8qc_111[b,mu,b,mu] := -4600*z2*z3*z7-2252*z2*z5^2-11248*z6*z3^2-20424*z4*
z3*z5+144*z4*Zeta53+424*z2*Zeta73+68988*z3*z9+194000/3*z5*z7+22996/9*Zeta93
-75111271853/199008*z12: EZMHVg8qc_111[b,mu,mu,mv] := 2624*z2*z3*z7+1268*z2*z5^2
+6371*z6*z3^2+12012*z4*z3*z5-156*z4*Zeta53-288*z2*Zeta73-39624*z3*z9-115960/3*
z5*z7-13208/9*Zeta93+86671104197/398016*z12: EZMHVg8qc_111[b,mu,mv,mv] := 0: 
EZMHVg8qc_111[b,mu,mw,mu] := 0: EZMHVg8qc_111[b,mw,b,mu] := -4600*z2*z3*z7-2252
*z2*z5^2-11248*z6*z3^2-20424*z4*z3*z5+144*z4*Zeta53+424*z2*Zeta73+68988*z3*z9+
194000/3*z5*z7+22996/9*Zeta93-75111271853/199008*z12:
EZMHVg8qc_111[b,mw,mw,mw] :=
-12363*z6*z3^2-2260*z2*z5^2+1500*z4*Zeta53+1312*z2*Zeta73-29676*z4*z3*z5
-5088*z2*z3*z7+105480*z5*z7+84312*z3*z9-4134176867/132672*z12+9368/3*Zeta93: 
EZMHVg8qc_111[b,yw,yw,mw] := -11543/2*z6*z3^2-1128*z2*z5^2+270*z4*Zeta53+276*z2
*Zeta73-10386*z4*z3*z5-2380*z2*z3*z7+105700/3*z5*z7+35322*z3*z9-190134065747/
398016*z12+11774/9*Zeta93: EZMHVg8qc_111[c,c,c,mu] := 30812661671/16584*z12+
104615/2*z6*z3^2+99210*z4*z3*z5-1566*z4*Zeta53+21364*z2*z3*z7+10316*z2*z5^2
-2508*z2*Zeta73-325710*z3*z9-322140*z5*z7-36190/3*Zeta93:
EZMHVg8qc_111[c,c,c,yv] := 0:
EZMHVg8qc_111[c,c,mu,mv] := 160553227/2764*z12+1494*z6*z3^2+3600*z4*z3
*z5-168*z4*Zeta53+648*z2*z3*z7+284*z2*z5^2-152*z2*Zeta73-10260*z3*z9-12640*z5*
z7-380*Zeta93: EZMHVg8qc_111[c,mu,b,mu] := -5248*z2*z3*z7-2536*z2*z5^2-12742*z6
*z3^2-24024*z4*z3*z5+312*z4*Zeta53+576*z2*Zeta73+79248*z3*z9+231920/3*z5*z7+
26416/9*Zeta93-86671104197/199008*z12:
EZMHVg8qc_111[c,mu,c,mu] := -4600*z2*z3*z7-2252*z2*z5^2-11248*z6*z3^2
-20424*z4*z3*z5+144*z4*Zeta53+424*z2*Zeta73+68988*z3*z9+194000/3*z5*z7
+22996/9*Zeta93-75111271853/199008*z12:
EZMHVg8qc_111[c,mu,mu,mu] := -12363*z6*z3^2-2260*z2*z5^2+1500*z4*Zeta53
+1312*z2*Zeta73-29676*z4*z3*z5-5088*z2*z3*z7+105480*z5*z7+84312*z3*z9
-4134176867/132672*z12+9368/3*Zeta93:
EZMHVg8qc_111[c,mu,mv,mv] := 2624*z2*z3*z7+1268*z2*z5^2+6371*z6*z3^2
+12012*z4*z3*z5-156*z4*Zeta53-288*z2*Zeta73-39624*z3*z9-115960/3*z5*z7
-13208/9*Zeta93+86671104197/398016*z12:
EZMHVg8qc_111[c,mu,yw,mu] := 0: EZMHVg8qc_111[c,mu,yw,mv] := 0:
EZMHVg8qc_111[c,mu,yw,mw] := 0:
EZMHVg8qc_111[c,mv,c,mu] := -4600*z2*z3*z7-2252*z2*z5^2-11248*z6*z3^2
-20424*z4*z3*z5+144*z4*Zeta53+424*z2*Zeta73+68988*z3*z9+194000/3*z5*z7
+22996/9*Zeta93-75111271853/199008*z12: 
EZMHVg8qc_111[c,mv,c,yv] := 0: EZMHVg8qc_111[c,mv,mv,yw] := 0:
EZMHVg8qc_111[c,mv,mw,mu] := -2624*z2*z3*z7-1268*z2*z5^2-6371*z6*z3^2
-12012*z4*z3*z5+156*z4*Zeta53+288*z2*Zeta73+39624*z3*z9+115960/3*z5*z7
+13208/9*Zeta93-86671104197/398016*z12:
EZMHVg8qc_111[mu,b,b,mu] := 0: EZMHVg8qc_111[mu,b,mu,mu] := 0: 
EZMHVg8qc_111[mu,b,mw,yw] := 0: EZMHVg8qc_111[mu,c,yv,mv] := 0:
EZMHVg8qc_111[mu,mu,b,mu] := 0: EZMHVg8qc_111[mu,mu,c,mu] := 0:
EZMHVg8qc_111[mu,mu,mu,mu] := 0: EZMHVg8qc_111[mu,mu,mu,mv] := 0:
EZMHVg8qc_111[mu,mu,mv,mv] := 0: EZMHVg8qc_111[mu,mv,yw,mu] := 0:
EZMHVg8qc_111[mu,mv,yw,mv] := 0: EZMHVg8qc_111[mu,mv,yw,mw] := 0:
EZMHVg8qc_111[mu,mw,b,mu] := 0: EZMHVg8qc_111[mu,mw,yv,mv] := 0:
EZMHVg8qc_111[mu,mw,yw,mw] := 0: EZMHVg8qc_111[mu,yv,mv,yw] := 0:
EZMHVg8qc_111[mu,yv,mw,yw] := 0: EZMHVg8qc_111[mu,yv,yu,mw] := 0: 
EZMHVg8qc_111[mu,yv,yv,mv] := 0: EZMHVg8qc_111[mv,a,a,mv] := 0:
EZMHVg8qc_111[mv,a,mw,mw] := 0: EZMHVg8qc_111[mv,c,c,mu] := 0:
EZMHVg8qc_111[mv,c,c,yv] := 0: EZMHVg8qc_111[mv,c,mu,mv] := 0:
EZMHVg8qc_111[mv,c,yv,mv] := 0: EZMHVg8qc_111[mv,mu,c,yv] := 0:
EZMHVg8qc_111[mv,mu,mw,yw] := 0: EZMHVg8qc_111[mv,mv,mv,mv] := 0:
EZMHVg8qc_111[mv,mv,mv,mw] := 0: EZMHVg8qc_111[mv,mv,mw,mu] := 0: 
EZMHVg8qc_111[mv,mv,yw,mu] := 0: EZMHVg8qc_111[mv,mv,yw,mv] := 0: 
EZMHVg8qc_111[mv,mw,a,mv] := 0: EZMHVg8qc_111[mv,mw,b,mu] := 0:
EZMHVg8qc_111[mv,mw,mw,mw] := 0: EZMHVg8qc_111[mv,yu,mu,mv] := 0:
EZMHVg8qc_111[mv,yv,yv,mv] := 0: EZMHVg8qc_111[mv,yw,mu,mu] := 0:
EZMHVg8qc_111[mv,yw,mv,mw] := 0: EZMHVg8qc_111[mv,yw,mv,yw] := 0:
EZMHVg8qc_111[mv,yw,mw,mu] := 0: EZMHVg8qc_111[mv,yw,mw,mw] := 0:
EZMHVg8qc_111[mv,yw,yv,mv] := 0: EZMHVg8qc_111[mv,yw,yv,mw] := 0:
EZMHVg8qc_111[mv,yw,yw,mu] := 0: EZMHVg8qc_111[mv,yw,yw,mw] := 0:
EZMHVg8qc_111[mw,a,mv,yw] := 0: EZMHVg8qc_111[mw,a,yv,mv] := 0:
EZMHVg8qc_111[mw,a,yv,mw] := 0: EZMHVg8qc_111[mw,b,mw,yw] := 0:
EZMHVg8qc_111[mw,mu,mv,mv] := 0: EZMHVg8qc_111[mw,mu,mw,mu] := 0: 
EZMHVg8qc_111[mw,mu,yv,mw] := 0: EZMHVg8qc_111[mw,mu,yw,mu] := 0: 
EZMHVg8qc_111[mw,mu,yw,mw] := 0: EZMHVg8qc_111[mw,mv,a,mv] := 0:
EZMHVg8qc_111[mw,mv,c,mu] := 0: EZMHVg8qc_111[mw,mv,c,yv] := 0:
EZMHVg8qc_111[mw,mv,mu,mv] := 0: EZMHVg8qc_111[mw,mv,yu,mw] := 0:
EZMHVg8qc_111[mw,mw,mw,mw] := 0: EZMHVg8qc_111[mw,mw,yw,mw] := 0:
EZMHVg8qc_111[mw,yu,b,mu] := 0: EZMHVg8qc_111[mw,yu,mu,mu] := 0:
EZMHVg8qc_111[mw,yu,mu,mv] := 0: EZMHVg8qc_111[mw,yw,mu,mu] := 0:
EZMHVg8qc_111[mw,yw,mv,yw] := 0: EZMHVg8qc_111[mw,yw,yv,mv] := 0: 
EZMHVg8qc_111[mw,yw,yv,mw] := 0: EZMHVg8qc_111[mw,yw,yw,mu] := 0: 
EZMHVg8qc_111[mw,yw,yw,mw] := 0:
EZMHVg8qc_111[yu,a,mv,yw] := 1264*z2*z3*z7+472*z2*z5^2+3139*z6*z3^2
+8316*z4*z3*z5-948*z4*Zeta53-568*z2*Zeta73-22272*z3*z9-106240/3*z5*z7
-7424/9*Zeta93-2005458709/24876*z12:
EZMHVg8qc_111[yu,a,yv,mv] := 244*z2*z3*z7+140*z2*z5^2+1199/2*z6*z3^2
+1626*z4*z3*z5+114*z4*Zeta53-12*z2*Zeta73-4302*z3*z9-3420*z5*z7-478/3*Zeta93
-17243826925/66336*z12:
EZMHVg8qc_111[yu,a,yv,mw] := 244*z2*z3*z7+140*z2*z5^2+1199/2*z6*z3^2
+1626*z4*z3*z5+114*z4*Zeta53-12*z2*Zeta73-4302*z3*z9-3420*z5*z7-478/3*Zeta93
-17243826925/66336*z12: EZMHVg8qc_111[yu,b,mw,yw] := 1264*z2*z3*z7
+472*z2*z5^2+3139*z6*z3^2+8316*z4*z3*z5-948*z4*Zeta53-568*z2*Zeta73
-22272*z3*z9-106240/3*z5*z7-7424/9*Zeta93-2005458709/24876*z12:
EZMHVg8qc_111[yu,mu,mw,yw] := 2624*z2*z3*z7+1268*z2*z5^2+6371*z6*z3^2
+12012*z4*z3*z5-156*z4*Zeta53-288*z2*Zeta73-39624*z3*z9-115960/3*z5*z7
-13208/9*Zeta93+86671104197/398016*z12:
EZMHVg8qc_111[yu,mv,a,mv] := 0: EZMHVg8qc_111[yu,mv,c,mu] := 0:
EZMHVg8qc_111[yu,mv,mu,mv] := 0: EZMHVg8qc_111[yu,mv,mv,mv] := 0:
EZMHVg8qc_111[yu,mv,mv,mw] := 0:
EZMHVg8qc_111[yu,mw,yv,mv] := 2624*z2*z3*z7+1268*z2*z5^2+6371*z6*z3^2
+12012*z4*z3*z5-156*z4*Zeta53-288*z2*Zeta73-39624*z3*z9-115960/3*z5*z7
-13208/9*Zeta93+86671104197/398016*z12: 
EZMHVg8qc_111[yv,a,yv,mw] := 244*z2*z3*z7+140*z2*z5^2+1199/2*z6*z3^2
+1626*z4*z3*z5+114*z4*Zeta53-12*z2*Zeta73-4302*z3*z9-3420*z5*z7-478/3*Zeta93
-17243826925/66336*z12:
EZMHVg8qc_111[yv,b,b,mu] := 0:
EZMHVg8qc_111[yv,b,mw,yw] := 1264*z2*z3*z7+472*z2*z5^2+3139*z6*z3^2
+8316*z4*z3*z5-948*z4*Zeta53-568*z2*Zeta73-22272*z3*z9-106240/3*z5*z7
-7424/9*Zeta93-2005458709/24876*z12:
EZMHVg8qc_111[yv,c,c,yv] := -5079/2*z6*z3^2-332*z2*z5^2+1062*z4*Zeta53
+556*z2*Zeta73-6690*z4*z3*z5-1020*z2*z3*z7+95980/3*z5*z7+17970*z3*z9
-35687811103/199008*z12+5990/9*Zeta93: 
EZMHVg8qc_111[yv,c,yv,mv] := 244*z2*z3*z7+140*z2*z5^2+1199/2*z6*z3^2
+1626*z4*z3*z5+114*z4*Zeta53-12*z2*Zeta73-4302*z3*z9-3420*z5*z7-478/3*Zeta93
-17243826925/66336*z12:
EZMHVg8qc_111[yv,mu,c,yv] := 0: EZMHVg8qc_111[yv,mu,mw,mu] := 0: 
EZMHVg8qc_111[yv,mv,a,mv] := 0: EZMHVg8qc_111[yv,mv,c,mu] := 0:
EZMHVg8qc_111[yv,mv,mu,mv] := 0: EZMHVg8qc_111[yv,mv,mv,mv] := 0:
EZMHVg8qc_111[yw,a,mw,mw] := 0: EZMHVg8qc_111[yw,b,b,mu] := 0:
EZMHVg8qc_111[yw,b,mu,mu] := 0: EZMHVg8qc_111[yw,c,c,mu] := 0:
EZMHVg8qc_111[yw,c,mu,mv] := 0: EZMHVg8qc_111[yw,mu,b,mu] := 0:
EZMHVg8qc_111[yw,mu,c,mu] := 0: EZMHVg8qc_111[yw,mu,mu,mu] := 0:
EZMHVg8qc_111[yw,mu,mu,mv] := 0: EZMHVg8qc_111[yw,mu,mv,mv] := 0: 
EZMHVg8qc_111[yw,mu,mw,mu] := 0:
EZMHVg8qc_111[yw,mu,mw,yw] := 2624*z2*z3*z7+1268*z2*z5^2+6371*z6*z3^2
+12012*z4*z3*z5-156*z4*Zeta53-288*z2*Zeta73-39624*z3*z9-115960/3*z5*z7
-13208/9*Zeta93+86671104197/398016*z12:
EZMHVg8qc_111[yw,mu,yw,mu] := -7872*z2*z3*z7-3804*z2*z5^2-19113*z6*z3^2
-36036*z4*z3*z5+468*z4*Zeta53+864*z2*Zeta73+118872*z3*z9+115960*z5*z7
+13208/3*Zeta93-86671104197/132672*z12:
EZMHVg8qc_111[yw,mu,yw,mv] := 5248*z2*z3*z7+2536*z2*z5^2+12742*z6*z3^2
+24024*z4*z3*z5-312*z4*Zeta53-576*z2*Zeta73-79248*z3*z9-231920/3*z5*z7
-26416/9*Zeta93+86671104197/199008*z12:
EZMHVg8qc_111[yw,mu,yw,mw] := 2624*z2*z3*z7+1268*z2*z5^2+6371*z6*z3^2
+12012*z4*z3*z5-156*z4*Zeta53-288*z2*Zeta73-39624*z3*z9-115960/3*z5*z7
-13208/9*Zeta93+86671104197/398016*z12:
EZMHVg8qc_111[yw,mv,c,yv] := 0: EZMHVg8qc_111[yw,mv,mv,mw] := 0:
EZMHVg8qc_111[yw,mv,mv,yw] := -7872*z2*z3*z7-3804*z2*z5^2-19113*z6*z3^2
-36036*z4*z3*z5+468*z4*Zeta53+864*z2*Zeta73+118872*z3*z9+115960*z5*z7
+13208/3*Zeta93-86671104197/132672*z12:
EZMHVg8qc_111[yw,mv,mw,mu] := 0: EZMHVg8qc_111[yw,mw,a,mv] := 0:
EZMHVg8qc_111[yw,mw,b,mu] := 0: EZMHVg8qc_111[yw,mw,mw,mw] := 0:
EZMHVg8qc_111[yw,yv,c,yv] := 0: EZMHVg8qc_111[yw,yv,mv,yw] := 0:
EZMHVg8qc_111[yw,yv,mw,yw] := 0: EZMHVg8qc_111[yw,yv,yu,mw] := 0:
EZMHVg8qc_111[yw,yv,yv,mv] := 0: EZMHVg8qc_111[yw,yw,mv,yw] := 0: 
EZMHVg8qc_111[yw,yw,yv,mv] := 0: EZMHVg8qc_111[yw,yw,yv,mw] := 0: 
EZMHVg8qc_111[yw,yw,yw,mu] := 0: EZMHVg8qc_111[yw,yw,yw,mw] := 0: 
EZMHVg8qc_111[yw,yw,yw,yu] := -38024*z2*z3*z7-15860*z2*z5^2-97748*z6*z3^2
-194952*z4*z3*z5+20448*z4*Zeta53+11224*z2*Zeta73+616644*z3*z9+830480*z5*z7
+68516/3*Zeta93-701035576445/132672*z12:
EZMHVg8qc_111[yw,yw,yw,yv] := -38024*z2*z3*z7-15860*z2*z5^2-97748*z6*z3^2
-194952*z4*z3*z5+20448*z4*Zeta53+11224*z2*Zeta73+616644*z3*z9+830480*z5*z7
+68516/3*Zeta93-701035576445/132672*z12: 
EZMHVg8qc_111[yw,yw,yw,yw] := -113873*z6*z3^2-19096*z2*z5^2+20580*z4*Zeta53
+11784*z2*Zeta73-223788*z4*z3*z5-44600*z2*z3*z7+921160*z5*z7+714996*z3*z9
-386146785425/66336*z12+79444/3*Zeta93:
#
# The remaining quadruples can be found using "Equadruple", or the relations
# "quadrels392" in "MHV8quintuples.txt".
#
# The 8 loop quintuples are weight 11, so we don't need to specify them here.
#
############################################################
############################################################
# REPRISE ALL RESULTS IN THE f-ALPHABET.
#
# Here f() = 1.   f(i1,i2,...) = f_{i1,i2,...} in the paper.
#
#########################################################
# First we give the bases for Hhex(1,1,1), through weight 12,
# where we know the dropouts:
#
ZetafBasis[1] := [ ] :
ZetafBasis[2] := [ z2*f() ]:
ZetafBasis[3] := [ ] :                          # 1 dropout
ZetafBasis[4] := [ z4*f() ] :
ZetafBasis[5] := [ 5*f(5)-2*z2*f(3) ] :         # 1 dropout
ZetafBasis[6] := [ z6*f() ] :                   # 
ZetafBasis[7] := [ 7*f(7)-z2*f(5)-3*z4*f(3) ] : # 2 dropouts
ZetafBasis[8] := [ z8*f(), 5*f(3,5)-2*z2*f(3,3) ] :
#
ZetafBasis[9] := [ 7*f(9)-6*z4*f(5), 5*f(9)-3*z6*f(3),
                   z2*f(7)-z6*f(3) ] :          # 1 dropout
#
ZetafBasis[10] := [ z10*f(), 7*f(3,7)-z2*f(3,5)-3*z4*f(3,3), 
            5*f(5,5)-2*z2*f(5,3) ] :
#
ZetafBasis[11] := [ 33*f(11)-20*z8*f(3), z2*f(9)-z8*f(3),
            3*z4*f(7)-2*z8*f(3), 3*z6*f(5)-2*z8*f(3),
            5*f(3,3,5)-2*z2*f(3,3,3)+5611/132*z8*f(3) ] :  # 1 dropout
#
ZetafBasis[12] := [ z12*f(),
7*f(3,9)-6*z4*f(3,5) + 1/3 * (7*f(5,7)-z2*f(5,5)-3*z4*f(5,3)),
7 * (5*f(3,9)-3*z6*f(3,3)) + 5/3 * (7*f(5,7)-z2*f(5,5)-3*z4*f(5,3)),
z2*f(3,7)-z6*f(3,3),
3 * (5*f(3,9)-3*z6*f(3,3)) - (5*f(7,5)-2*z2*f(7,3)) ] :  # 1 dropout
#
# Beyond weight 13, we expect dropouts, but we can't determine them yet,
# so these bases are likely over-complete.
#
ZetafBasis[13] := [ f(13),
  z10*f(3), 7*f(3,3,7)-z2*f(3,3,5)-3*z4*f(3,3,3), 5*f(3,5,5)-2*z2*f(3,5,3),
  z8*f(5), 5*f(5,3,5)-2*z2*f(5,3,3),
  z6*f(7),
  z4*f(9),
  z2*f(11) ] :
#
# Even at 8-loops-MHV, we only see 1 number (see below).
# It does lie in this space.
#
ZetafBasis[14] := [ z14*f(),
  33*f(3,11)-20*z8*f(3,3), z2*f(3,9)-z8*f(3,3),
  3*z4*f(3,7)-2*z8*f(3,3), 3*z6*f(3,5)-2*z8*f(3,3),
  5*f(3,3,3,5)-2*z2*f(3,3,3,3)+5611/132*z8*f(3,3),
  7*f(5,9)-6*z4*f(5,5), 5*f(5,9)-3*z6*f(5,3), z2*f(5,7)-z6*f(5,3),
  7*f(7,7)-z2*f(7,5)-3*z4*f(7,3),
  5*f(9,5)-2*z2*f(9,3) ] :
#
# There are 14 possible elements of ZetafBasis[15]
# and 20 possible elements of ZetafBasis[16].
#
# But it is simpler to just act with the \del_{2k+1} derivations
# on EZMHVg8_111 below, and check that the results are in ZetafBasis[13],
# ZetafBasis[11], etc.
#
###############################################################
# The full MHV amplitude at (1,1,1) through 8 loops.
#
EZMHVfg0_111 := f() :
EZMHVfg1_111 := 0 :
EZMHVfg2_111 := - 9 * z4*f() :
EZMHVfg3_111 := 121 * z6*f() :
EZMHVfg4_111 := - 6381/4 * z8*f() + 24 * (5*f(3,5)-2*z2*f(3,3)) :
#
EZMHVfg5_111 := 222069/10*z10*f()
- 384 * (7*f(3,7)-z2*f(3,5)-3*z4*f(3,3))
- 312 * (5*f(5,5)-2*z2*f(5,3)) :
#
EZMHVfg6_111 := - 10709056285/33168 * z12*f()
+ 2244 * (7*f(3,9)-6*z4*f(3,5))
+ 6564 * (5*f(3,9)-3*z6*f(3,3))
- 3072 * (z2*f(3,7)-z6*f(3,3))
+ 5328 * (7*f(5,7)-z2*f(5,5)-3*z4*f(5,3))
+ 4224 * (5*f(7,5)-2*z2*f(7,3)) :
#
EZMHVfg7_111 := 7113137749/5040 * z14*f() - 63968 * (5*f(9,5)-2*z2*f(9,3))
- 77952 * (7*f(7,7)-z2*f(7,5)-3*z4*f(7,3))
- 34716 * (7*f(5,9)-6*z4*f(5,5))
- 95916 * (5*f(5,9)-3*z6*f(5,3))
+ 44640 * (z2*f(5,7)-z6*f(5,3))
- 413920/11 * (33*f(3,11)-20*z8*f(3,3))
+ 28000 * (z2*f(3,9)-z8*f(3,3))
+ 61824 * (3*z4*f(3,7)-2*z8*f(3,3))
+ 72456 * (3*z6*f(3,5)-2*z8*f(3,3))
- 4992 * (5*f(3,3,3,5)-2*z2*f(3,3,3,3)+5611/132*z8*f(3,3)) :
#
EZMHVfg8_111 := 
9122624*f(9,7)+11543472*f(7,9)+5153280*f(11,5)+19603536*f(5,11)
+23915376*f(3,13)+371520*f(5,3,3,5)+400320*f(3,3,5,5)+400320*f(3,5,3,5)
+825216*f(3,3,3,7)
+ z2 * (-701856*f(7,7)-1303232*f(9,5)-430656*f(5,9)
        -2061312*f(11,3)+309696*f(3,11)
        -160128*f(3,5,3,3)-160128*f(3,3,5,3)
        -117888*f(3,3,3,5)-148608*f(5,3,3,3))
+ z4 * (-3243888*f(5,7)-3475296*f(7,5)-3909696*f(9,3)-3215472*f(3,9)
        -353664*f(3,3,3,3))
+ z6 * (-3612804*f(5,5)-3791520*f(7,3)-3409152*f(3,7))
+ z8 * (-3720664*f(5,3)-3456614*f(3,5))
-19560489/5 * z10 * f(3,3)
-512193667550809/7639104 * z16 * f() :
#
# Next we give just the nonvanishing 7-loop double coproducts
# (up to dihedral images):
#
EZMHVfg7dc_111[a,mv] := -41184375511/199008*z12*f()
+29616*f(3,9)+21952*f(5,7)+12080*f(7,5)
-1920*z2*f(3,7)-3136*z2*f(5,5)-4832*z2*f(7,3)
-9048*z4*f(3,5)-9408*z4*f(5,3)-9516*z6*f(3,3) :
#
EZMHVfg7dc_111[yw,yu] := 85695597061/49752*z12*f()
-168432*f(3,9)-158704*f(5,7)-102560*f(7,5)
+10560*z2*f(3,7)+22672*z2*f(5,5)+41024*z2*f(7,3)
+51096*z4*f(3,5)+68016*z4*f(5,3)+54732*z6*f(3,3) :
#
# One can check that these are both linear combinations of the
# 5 basis elements in ZetafBasis[12].
#
# Next we give just the nonvanishing 8-loop double coproducts
# (up to dihedral images):
#
EZMHVfg8dc_111[a,mv] := -42437345879/60480*z14*f()
-1123408*f(3,11)-593136*f(5,9)-456064*f(7,7)-827840/3*f(9,5)-28160*f(3,3,3,5)
+75680/3*z2*f(3,9)+36768*z2*f(5,7)+65152*z2*f(7,5)+331136/3*z2*f(9,3)
+11264*z2*f(3,3,3,3)
+180336*z4*f(3,7)+195168*z4*f(5,5)+195456*z4*f(7,3)
+185116*z6*f(3,5)+182496*z6*f(5,3)+517768/3*z8*f(3,3) :
#
EZMHVfg8dc_111[yw,yu] := -323971645187/30240*z14*f()
+5593216*f(3,11)+3881664*f(5,9)+3385984*f(7,7)+6522080/3*f(9,5)
+122240*f(3,3,3,5)
-374816/3*z2*f(3,9)-235488*z2*f(5,7)-483712*z2*f(7,5)-2608832/3*z2*f(9,3)
-48896*z2*f(3,3,3,3)
-955200*z4*f(3,7)-1249752*z4*f(5,5)-1451136*z4*f(7,3)
-993784*z6*f(3,5)-1218684*z6*f(5,3)-2779024/3*z8*f(3,3) :
#
# One can check that these are both linear combinations of the
# 11 basis elements in ZetafBasis[14].
#
#
# Next we give the one independent weight 13 triple coproduct at (1,1,1)
# at 8 loops:
#
EZMHVfg8tc_111[a,a,mv] :=
-996474*f(13)-34384*f(3,3,7)-16680*f(3,5,5)-16680*f(5,3,5)
-12904*z2*f(11)+4912*z2*f(3,3,5)+6672*z2*f(3,5,3)+6672*z2*f(5,3,3)
+133978*z4*f(9)+14736*z4*f(3,3,3)
+142048*z6*f(7)+1728307/12*z8*f(5)+6520163/40*z10*f(3) :
#
# One can check that this is a linear combination of the
# 9 basis elements in ZetafBasis[13].
#
###################################
# 8 loop Quadruples:
#
# We provide 5 linearly independent ones in the f-alphabet.
# The other nonzero ones are simple linear combinations of these,
# which can be worked out from the conventional representation given above.
#
EZMHVfg8qc_111[a,a,a,mv] := 30812661671/16584*z12*f()
+104615*z6*f(3,3)+99210*z4*f(3,5)+107040*z4*f(5,3)+21364*z2*f(3,7)
+56476*z2*f(7,3)+35680*z2*f(5,5)-249760*f(5,7)-141190*f(7,5)-325710*f(3,9):
#
EZMHVfg8qc_111[a,a,mw,mw] := 160553227/2764*z12*f()
+2988*z6*f(3,3)+3600*z4*f(3,5)+4440*z4*f(5,3)+648*z2*f(3,7)+2776*z2*f(7,3)
+1480*z2*f(5,5)-10360*f(5,7)-6940*f(7,5)-10260*f(3,9):
#
EZMHVfg8qc_111[a,mv,mu,mv] := 86671104197/398016*z12*f()
+2624*z2*f(3,7)+6656*z2*f(7,3)+4264*z2*f(5,5)+12742*z6*f(3,3)
+12012*z4*f(3,5)+12792*z4*f(5,3)-39624*f(3,9)-29848*f(5,7)-16640*f(7,5):
#
EZMHVfg8qc_111[a,mv,mv,mv] := -4134176867/132672*z12*f()
-24726*z6*f(3,3)-12392*z2*f(5,5)-37176*z4*f(5,3)-23456*z2*f(7,3)
-29676*z4*f(3,5)-5088*z2*f(3,7)+86744*f(5,7)+58640*f(7,5)+84312*f(3,9):
#
EZMHVfg8qc_111[b,yw,yw,mw] := -190134065747/398016*z12*f()
-11543*z6*f(3,3)-3912*z2*f(5,5)-11736*z4*f(5,3)-6244*z2*f(7,3)
-10386*z4*f(3,5)-2380*z2*f(3,7)+27384*f(5,7)+15610*f(7,5)+35322*f(3,9):
#
# These all belong to ZetafBasis[12].
#