Extended Griess Algebras and Matsuo-Norton Trace Formulae

Abstract

We introduce the $\mathbb{Z}_{2}$-extended Griess algebra of a vertex operator superalgebra with an involution and derive the Matsuo-Norton trace formulae for the extended Griess algebra based on conformal design structure. We illustrate an application of our formulae by reformulating the one-to-one correspondence between 2A-elements of the Baby-monster simple group and N = 1 c = 7∕10 Virasoro subalgebras inside the Baby-monster vertex operator superalgebra.

Appendix

1.1 Coefficients in Generalized Casimir Vectors

$A_{[2]}^{(2)} = 2 {\ast} h {\ast} d$, $A_{[3]}^{(3)} = h {\ast} d$, $A_{[4]}^{(4)} = 3 {\ast} h {\ast} d {\ast} (c - 2 {\ast} h + 4)$, $A_{[2,2]}^{(4)} = 2 {\ast} h {\ast} (5 {\ast} h + 1) {\ast} d$, $A_{[5]}^{(5)} = 2 {\ast} h {\ast} d {\ast} (c - 2 {\ast} h + 4)$, $A_{[3,2]}^{(5)} = 2 {\ast} h {\ast} (5 {\ast} h + 1) {\ast} d$, $A_{[6]}^{(6)} = 4{\ast}h{\ast}d{\ast}(5{\ast}c^{3}+(-15{\ast}h+65){\ast}c^{2}+(-20{\ast}h^{2}-148{\ast}h+148){\ast}c-26{\ast}h^{2}+98{\ast}h-92)$, $A_{[4,2]}^{(6)} = 2{\ast}h{\ast}d{\ast}((42{\ast}h+8){\ast}c^{2} +(-84{\ast}h^{2} +349{\ast}h+65){\ast}c-134{\ast}h^{2} -86{\ast}h-40)$, $A_{[3,3]}^{(6)} = (1/2){\ast}h{\ast}d{\ast}((70{\ast}h+15){\ast}c^{2} +(614{\ast}h+136){\ast}c+248{\ast}h^{2} -464{\ast}h-64)$, $A_{[2,2,2]}^{(6)} = (4/3) {\ast} h {\ast} d {\ast} ((70 {\ast} h^{2} + 42 {\ast} h + 8) {\ast} c + 29 {\ast} h^{2} - 57 {\ast} h - 2)$, $A_{[7]}^{(7)} = 3{\ast}h{\ast}d{\ast}(5{\ast}c^{3}+(-15{\ast}h+65){\ast}c^{2}+(-20{\ast}h^{2}-148{\ast}h+148){\ast}c-26{\ast}h^{2}+98{\ast}h-92)$, $A_{[5,2]}^{(7)} = 2{\ast}h{\ast}d{\ast}((28{\ast}h+5){\ast}c^{2} +(-56{\ast}h^{2} +243{\ast}h+41){\ast}c-172{\ast}h^{2} -16{\ast}h-28)$, $A_{[4,3]}^{(7)} = 3{\ast}h{\ast}d{\ast}((14{\ast}h+3){\ast}c^{2} +(-28{\ast}h^{2} +106{\ast}h+24){\ast}c+38{\ast}h^{2} -70{\ast}h-12)$, $A_{[3,2,2]}^{(7)} = 2 {\ast} h {\ast} d {\ast} ((70 {\ast} h^{2} + 42 {\ast} h + 8) {\ast} c + 29 {\ast} h^{2} - 57 {\ast} h - 2)$, $A_{[8]}^{(8)} = (1/2){\ast}h{\ast}d{\ast}(350{\ast}c^{5}+(-1260{\ast}h+10080){\ast}c^{4}+(-560{\ast}h^{2}-31735{\ast}h+85005){\ast}c^{3}+(-5040{\ast}h^{3}-17240{\ast}h^{2}-192290{\ast}h+194494){\ast}c^{2}+(-18520{\ast}h^{3}-43840{\ast}h^{2}+20928{\ast}h-8184){\ast}c+4344{\ast}h^{3}-32496{\ast}h^{2}+76488{\ast}h-57744)$, A _[6, 2] ⁽⁸⁾ = 2∗h∗d∗((300∗h+50)∗c ⁴+(−900∗h ²+7312∗h+1176)∗c ³+(−1200∗h ³−18548∗h ²+42969∗h+6081)∗c ²+(−4552∗h ³−52960∗h ²+32406∗h−1466)∗c−536∗h ³−26880∗h ²+5696∗h−2808), $A_{[5,3]}^{(8)} = (1/2){\ast}h{\ast}d{\ast}((840{\ast}h+175){\ast}c^{4}+(-1680{\ast}h^{2}+19885{\ast}h+4188){\ast}c^{3}+(-25392{\ast}h^{2}+107936{\ast}h+23184){\ast}c^{2}+(-2016{\ast}h^{3}+1832{\ast}h^{2}+4060{\ast}h+968){\ast}c+7792{\ast}h^{3}+1776{\ast}h^{2}-31312{\ast}h-6816)$, $A_{[4,4]}^{(8)} = (3/2){\ast}h{\ast}d{\ast}((126{\ast}h+28){\ast}c^{4}+(-504{\ast}h^{2}+2787{\ast}h+643){\ast}c^{3}+(504{\ast}h^{3}-7156{\ast}h^{2}+13198{\ast}h+3338){\ast}c^{2}+(3180{\ast}h^{3}+2372{\ast}h^{2}-2480{\ast}h+344){\ast}c-2004{\ast}h^{3}+7248{\ast}h^{2}-6036{\ast}h-888)$, $A_{[4,2,2]}^{(8)} = 2{\ast}h{\ast}d{\ast}((630{\ast}h^{2}+366{\ast}h+68){\ast}c^{3}+(-1260{\ast}h^{3}+9159{\ast}h^{2}+4793{\ast}h+958){\ast}c^{2}+(-6942{\ast}h^{3}+11417{\ast}h^{2}-3187{\ast}h+210){\ast}c+1114{\ast}h^{3}-654{\ast}h^{2}-3064{\ast}h-168)$, $A_{[3,3,2]}^{(8)} = h{\ast}d{\ast}((1050{\ast}h^{2}+645{\ast}h+125){\ast}c^{3}+(16700{\ast}h^{2}+9170{\ast}h+1934){\ast}c^{2}+(3720{\ast}h^{3}+15510{\ast}h^{2}-8662{\ast}h+716){\ast}c-1016{\ast}h^{3}+6444{\ast}h^{2}-8692{\ast}h-264)$, $A_{[2,2,2,2]}^{(8)} = (2/3){\ast}h{\ast}d{\ast}((1050{\ast}h^{3}+1260{\ast}h^{2}+606{\ast}h+108){\ast}c^{2}+(3305{\ast}h^{3}-498{\ast}h^{2}-701{\ast}h+78){\ast}c-251{\ast}h^{3}+918{\ast}h^{2}-829{\ast}h-6)$, $A_{[9]}^{(9)} = (2/3){\ast}h{\ast}d{\ast}(210{\ast}c^{5}+(-756{\ast}h+6048){\ast}c^{4}+(-756{\ast}h^{2}-19311{\ast}h+50949){\ast}c^{3}+(-5544{\ast}h^{3}-19676{\ast}h^{2}-120486{\ast}h+115622){\ast}c^{2}+(-23508{\ast}h^{3}-30448{\ast}h^{2}+18372{\ast}h-5456){\ast}c+4428{\ast}h^{3}-26232{\ast}h^{2}+52020{\ast}h-34536)$, $A_{[7,2]}^{(9)} = 2{\ast}h{\ast}d{\ast}((225{\ast}h+35){\ast}c^{4}+(-675{\ast}h^{2}+5565{\ast}h+826){\ast}c^{3}+(-900{\ast}h^{3}-14615{\ast}h^{2}+33776{\ast}h+4267){\ast}c^{2}+(-2910{\ast}h^{3}-49778{\ast}h^{2}+29666{\ast}h-1378){\ast}c-2350{\ast}h^{3}-23916{\ast}h^{2}+6718{\ast}h-2196)$, $A_{[6,3]}^{(9)} = 4{\ast}h{\ast}d{\ast}((75{\ast}h+15){\ast}c^{4}+(-225{\ast}h^{2}+1747{\ast}h+350){\ast}c^{3}+(-300{\ast}h^{3}-3933{\ast}h^{2}+9193{\ast}h+1814){\ast}c^{2}+(-1642{\ast}h^{3}-3182{\ast}h^{2}+2740{\ast}h-88){\ast}c+1814{\ast}h^{3}-2964{\ast}h^{2}-1022{\ast}h-612)$, $A_{[5,4]}^{(9)} = 2{\ast}h{\ast}d{\ast}((126{\ast}h+28){\ast}c^{4}+(-504{\ast}h^{2}+2787{\ast}h+643){\ast}c^{3}+(504{\ast}h^{3}-7156{\ast}h^{2}+13198{\ast}h+3338){\ast}c^{2}+(3180{\ast}h^{3}+2372{\ast}h^{2}-2480{\ast}h+344){\ast}c-2004{\ast}h^{3}+7248{\ast}h^{2}-6036{\ast}h-888)$, $A_{[5,2,2]}^{(9)} = 4{\ast}h{\ast}d{\ast}((210{\ast}h^{2}+117{\ast}h+21){\ast}c^{3}+(-420{\ast}h^{3}+3208{\ast}h^{2}+1602{\ast}h+302){\ast}c^{2}+(-3554{\ast}h^{3}+4166{\ast}h^{2}-784{\ast}h+40){\ast}c+710{\ast}h^{3}-1032{\ast}h^{2}-746{\ast}h-60)$, $A_{[4,3,2]}^{(9)} = 2{\ast}h{\ast}d{\ast}((630{\ast}h^{2}+381{\ast}h+73){\ast}c^{3}+(-1260{\ast}h^{3}+8694{\ast}h^{2}+4780{\ast}h+1010){\ast}c^{2}+(-3222{\ast}h^{3}+10336{\ast}h^{2}-4022{\ast}h+300){\ast}c+98{\ast}h^{3}+1788{\ast}h^{2}-3890{\ast}h-156)$, $A_{[3,3,3]}^{(9)} = (1/6){\ast}h{\ast}d{\ast}((1050{\ast}h^{2}+675{\ast}h+135){\ast}c^{3}+(15770{\ast}h^{2}+9144{\ast}h+2038){\ast}c^{2}+(11160{\ast}h^{3}+13348{\ast}h^{2}-10332{\ast}h+896){\ast}c-3048{\ast}h^{3}+11328{\ast}h^{2}-10344{\ast}h-240)$, $A_{[3,2,2,2]}^{(9)} = (4/3){\ast}h{\ast}d{\ast}((1050{\ast}h^{3}+1260{\ast}h^{2}+606{\ast}h+108){\ast}c^{2}+(3305{\ast}h^{3}-498{\ast}h^{2}-701{\ast}h+78){\ast}c-251{\ast}h^{3}+918{\ast}h^{2}-829{\ast}h-6)$, $A_{[10]}^{(10)} = (6/5){\ast}h{\ast}d{\ast}(1050{\ast}c^{6}+(-4200{\ast}h+52290){\ast}c^{5}+(-3150{\ast}h^{2}-195019{\ast}h+888199){\ast}c^{4}+(-31500{\ast}h^{3}-160243{\ast}h^{2}-2900235{\ast}h+5888368){\ast}c^{3}+(-33600{\ast}h^{4}-876400{\ast}h^{3}-2224448{\ast}h^{2}-13733560{\ast}h+11872408){\ast}c^{2}+(-189616{\ast}h^{4}-3013900{\ast}h^{3}-3958988{\ast}h^{2}+2767600{\ast}h-800016){\ast}c-29792{\ast}h^{4}+816800{\ast}h^{3}-3744448{\ast}h^{2}+6247744{\ast}h-3575424)$, A _[8, 2] ⁽¹⁰⁾ = (1∕5)∗h∗d∗((19250∗h+2800)∗c ⁵+(−69300∗h ²+881440∗h+124040)∗c ⁴+(−30800∗h ³−2898185∗h ²+12963179∗h+1696856)∗c ³+(−277200∗h ⁴−1275240∗h ³−35996682∗h ²+64729982∗h+6705400)∗c ²+(−2026600∗h ⁴−3142080∗h ³−112130808∗h ²+64543216∗h−3235248)∗c−335864∗h ⁴−8601520∗h ³−49036936∗h ²+17453488∗h−4052928), $A_{[7,3]}^{(10)} = (3/10){\ast}h{\ast}d{\ast}((8250{\ast}h+1575){\ast}c^{5}+(-24750{\ast}h^{2}+368615{\ast}h+70030){\ast}c^{4}+(-33000{\ast}h^{3}-978660{\ast}h^{2}+5161264{\ast}h+966596){\ast}c^{3}+(-814640{\ast}h^{3}-10273412{\ast}h^{2}+22761712{\ast}h+3992640){\ast}c^{2}+(-103200{\ast}h^{4}-2524680{\ast}h^{3}-14390328{\ast}h^{2}+9889536{\ast}h-487008){\ast}c+139456{\ast}h^{4}+5842880{\ast}h^{3}-11650816{\ast}h^{2}-447872{\ast}h-1475328)$, $A_{[6,4]}^{(10)} = (12/5){\ast}h{\ast}d{\ast}((825{\ast}h+180){\ast}c^{5}+(-4125{\ast}h^{2}+35248{\ast}h+7832){\ast}c^{4}+(1650{\ast}h^{3}-149899{\ast}h^{2}+456809{\ast}h+104870){\ast}c^{3}+(6600{\ast}h^{4}+60200{\ast}h^{3}-1328561{\ast}h^{2}+1698137{\ast}h+418354){\ast}c^{2}+(67132{\ast}h^{4}+339830{\ast}h^{3}-202982{\ast}h^{2}-47264{\ast}h+18184){\ast}c-33620{\ast}h^{4}-77560{\ast}h^{3}+609380{\ast}h^{2}-579320{\ast}h-120480)$, $A_{[6,2,2]}^{(10)} = (4/5){\ast}h{\ast}d{\ast}((8250{\ast}h^{2}+4400{\ast}h+760){\ast}c^{4}+(-24750{\ast}h^{3}+296210{\ast}h^{2}+151096{\ast}h+26364){\ast}c^{3}+(-33000{\ast}h^{4}-801290{\ast}h^{3}+2704347{\ast}h^{2}+1191343{\ast}h+213790){\ast}c^{2}+(-232460{\ast}h^{4}-4589320{\ast}h^{3}+3842818{\ast}h^{2}-286646{\ast}h-11132){\ast}c+28644{\ast}h^{4}+857640{\ast}h^{3}-1710804{\ast}h^{2}-212088{\ast}h-52032)$, $A_{[5,5]}^{(10)} = (1/5){\ast}h{\ast}d{\ast}((4620{\ast}h+1050){\ast}c^{5}+(-18480{\ast}h^{2}+198719{\ast}h+46201){\ast}c^{4}+(18480{\ast}h^{3}-639607{\ast}h^{2}+2606009{\ast}h+632228){\ast}c^{3}+(443400{\ast}h^{3}-5076574{\ast}h^{2}+9773262{\ast}h+2648692){\ast}c^{2}+(41376{\ast}h^{4}+1074560{\ast}h^{3}+4481200{\ast}h^{2}-3129584{\ast}h+288608){\ast}c+293648{\ast}h^{4}-2776160{\ast}h^{3}+7129072{\ast}h^{2}-5166496{\ast}h-672384)$, $A_{[5,3,2]}^{(10)} = (1/5){\ast}h{\ast}d{\ast}((46200{\ast}h^{2}+27115{\ast}h+5075){\ast}c^{4}+(-92400{\ast}h^{3}+1636765{\ast}h^{2}+922753{\ast}h+178072){\ast}c^{3}+(-2371200{\ast}h^{3}+14798146{\ast}h^{2}+7158654{\ast}h+1515620){\ast}c^{2}+(-110880{\ast}h^{4}-9063800{\ast}h^{3}+17179184{\ast}h^{2}-5440168{\ast}h+338704){\ast}c-319088{\ast}h^{4}+2238560{\ast}h^{3}-652432{\ast}h^{2}-4969184{\ast}h-254976)$, $A_{[4,4,2]}^{(10)} = (3/5){\ast}h{\ast}d{\ast}((6930{\ast}h^{2}+4180{\ast}h+800){\ast}c^{4}+(-27720{\ast}h^{3}+226185{\ast}h^{2}+133283{\ast}h+26662){\ast}c^{3}+(27720{\ast}h^{4}-631060{\ast}h^{3}+1836386{\ast}h^{2}+928514{\ast}h+211140){\ast}c^{2}+(337140{\ast}h^{4}-1677780{\ast}h^{3}+2472564{\ast}h^{2}-812508{\ast}h+53544){\ast}c+100532{\ast}h^{4}-226520{\ast}h^{3}+432268{\ast}h^{2}-745384{\ast}h-34656)$, $A_{[4,3,3]}^{(10)} = (3/10){\ast}h{\ast}d{\ast}((11550{\ast}h^{2}+7425{\ast}h+1485){\ast}c^{4}+(-23100{\ast}h^{3}+389020{\ast}h^{2}+243566{\ast}h+51294){\ast}c^{3}+(-426920{\ast}h^{3}+3236462{\ast}h^{2}+1740898{\ast}h+429980){\ast}c^{2}+(-81840{\ast}h^{4}+1025620{\ast}h^{3}+3455108{\ast}h^{2}-2154976{\ast}h+167248){\ast}c-17296{\ast}h^{4}-522560{\ast}h^{3}+2125936{\ast}h^{2}-1996768{\ast}h-52992)$, $A_{[4,2,2,2]}^{(10)} = 4{\ast}h{\ast}d{\ast}((2310{\ast}h^{3}+2706{\ast}h^{2}+1276{\ast}h+224){\ast}c^{3}+(-4620{\ast}h^{4}+48797{\ast}h^{3}+50252{\ast}h^{2}+22925{\ast}h+4434){\ast}c^{2}+(-47038{\ast}h^{4}+140169{\ast}h^{3}-6264{\ast}h^{2}-27525{\ast}h+2578){\ast}c-4966{\ast}h^{4}+9340{\ast}h^{3}+15382{\ast}h^{2}-28252{\ast}h-288)$, $A_{[3,3,2,2]}^{(10)} = h{\ast}d{\ast}((11550{\ast}h^{3}+14025{\ast}h^{2}+6809{\ast}h+1222){\ast}c^{3}+(274840{\ast}h^{3}+284503{\ast}h^{2}+133429{\ast}h+26346){\ast}c^{2}+(40920{\ast}h^{4}+764986{\ast}h^{3}-116882{\ast}h^{2}-177916{\ast}h+18992){\ast}c+8648{\ast}h^{4}-92288{\ast}h^{3}+251080{\ast}h^{2}-201520{\ast}h-1344)$, $A_{[2,2,2,2,2]}^{(10)} = (4/15){\ast}h{\ast}d{\ast}((11550{\ast}h^{4}+23100{\ast}h^{3}+20130{\ast}h^{2}+8580{\ast}h+1440){\ast}c^{2}+(76675{\ast}h^{4}+30590{\ast}h^{3}-25615{\ast}h^{2}-10898{\ast}h+1608){\ast}c+3767{\ast}h^{4}-18410{\ast}h^{3}+29929{\ast}h^{2}-16342{\ast}h-24)$.

1.2 Coefficients in the Trace Formulae

$\mathrm{Sym}(a^{0}\vert \omega )(a^{1}\vert a^{2}) = (a^{0}\vert \omega )(a^{1}\vert a^{2}) + (a^{1}\vert \omega )(a^{0}\vert a^{2}) + (a^{2}\vert \omega )(a^{0}\vert a^{1})$, $\mathrm{Sym}(a^{0}\vert \omega )(a^{1}\vert \omega )(a^{2}\vert a^{3}) = (a^{0}\vert \omega )(a^{1}\vert \omega )(a^{2}\vert a^{3})+(a^{0}\vert \omega )(a^{2}\vert \omega )(a^{1}\vert a^{3})+(a^{0}\vert \omega )(a^{3}\vert \omega )(a^{1}\vert a^{2})+(a^{1}\vert \omega )(a^{2}\vert \omega )(a^{0}\vert a^{3}) + (a^{1}\vert \omega )(a^{3}\vert \omega )(a^{0}\vert a^{2}) + (a^{2}\vert \omega )(a^{3}\vert \omega )(a^{0}\vert a^{1})$, $\mathrm{Sym}(a^{0}\vert \omega )(a^{1}\vert a^{2}\vert a^{3}) = (a^{0}\vert \omega )(a^{1}\vert a^{2}\vert a^{3})+(a^{1}\vert \omega )(a^{0}\vert a^{2}\vert a^{3})+(a^{2}\vert \omega )(a^{0}\vert a^{1}\vert a^{3})+(a^{3}\vert \omega )(a^{0}\vert a^{1}\vert a^{2})$, $\mathrm{Sym}(a^{0}\vert a^{1})(a^{2}\vert a^{3}) = (a^{0}\vert a^{1})(a^{2}\vert a^{3}) + (a^{0}\vert a^{2})(a^{1}\vert a^{3}) + (a^{0}\vert a^{3})(a^{1}\vert a^{2})$, $\mathrm{Sym}(a^{0}\vert \omega )(a^{1}\vert \omega )(a^{2}\vert \omega )(a^{3}\vert a^{4}) = (a^{0}\vert \omega )(a^{1}\vert \omega )(a^{2}\vert \omega )(a^{3}\vert a^{4})+(a^{0}\vert \omega )(a^{1}\vert \omega )(a^{3}\vert \omega )(a^{2}\vert a^{4})+(a^{0}\vert \omega )(a^{1}\vert \omega )(a^{4}\vert \omega )(a^{2}\vert a^{3})+(a^{0}\vert \omega )(a^{2}\vert \omega )(a^{3}\vert \omega )(a^{1}\vert a^{4})+(a^{0}\vert \omega )(a^{2}\vert \omega )(a^{4}\vert \omega )(a^{1}\vert a^{3})+(a^{0}\vert \omega )(a^{3}\vert \omega )(a^{4}\vert \omega )(a^{1}\vert a^{2})+(a^{1}\vert \omega )(a^{2}\vert \omega )(a^{3}\vert \omega )(a^{0}\vert a^{4}) + (a^{1}\vert \omega )(a^{2}\vert \omega )(a^{4}\vert \omega )(a^{0}\vert a^{3}) + (a^{1}\vert \omega )(a^{3}\vert \omega )(a^{4}\vert \omega )(a^{0}\vert a^{2}) + (a^{2}\vert \omega )(a^{3}\vert \omega )(a^{4}\vert \omega )(a^{0}\vert a^{1})$, $\mathrm{Sym}(a^{0}\vert \omega )(a^{1}\vert \omega )(a^{2}\vert a^{3}\vert a^{4}) = (a^{0}\vert \omega )(a^{1}\vert \omega )(a^{2}\vert a^{3}\vert a^{4})+(a^{0}\vert \omega )(a^{2}\vert \omega )(a^{1}\vert a^{3}\vert a^{4})+(a^{0}\vert \omega )(a^{3}\vert \omega )(a^{1}\vert a^{2}\vert a^{4})+(a^{0}\vert \omega )(a^{4}\vert \omega )(a^{1}\vert a^{2}\vert a^{3})+(a^{1}\vert \omega )(a^{2}\vert \omega )(a^{0}\vert a^{3}\vert a^{4})+(a^{1}\vert \omega )(a^{3}\vert \omega )(a^{0}\vert a^{2}\vert a^{4})+(a^{1}\vert \omega )(a^{4}\vert \omega )(a^{0}\vert a^{2}\vert a^{3})+(a^{2}\vert \omega )(a^{3}\vert \omega )(a^{0}\vert a^{1}\vert a^{4}) + (a^{2}\vert \omega )(a^{4}\vert \omega )(a^{0}\vert a^{1}\vert a^{3}) + (a^{3}\vert \omega )(a^{4}\vert \omega )(a^{0}\vert a^{1}\vert a^{2})$, $\mathrm{Sym}(a^{0}\vert \omega )(a^{1}\vert a^{2})(a^{3}\vert a^{4}) = (a^{0}\vert \omega )(a^{1}\vert a^{2})(a^{3}\vert a^{4})+(a^{0}\vert \omega )(a^{1}\vert a^{3})(a^{2}\vert a^{4})+(a^{0}\vert \omega )(a^{1}\vert a^{4})(a^{2}\vert a^{3})+(a^{1}\vert \omega )(a^{0}\vert a^{2})(a^{3}\vert a^{4})+(a^{1}\vert \omega )(a^{0}\vert a^{3})(a^{2}\vert a^{4})+(a^{1}\vert \omega )(a^{0}\vert a^{4})(a^{2}\vert a^{3})+(a^{2}\vert \omega )(a^{0}\vert a^{1})(a^{3}\vert a^{4})+(a^{2}\vert \omega )(a^{0}\vert a^{3})(a^{1}\vert a^{4})+(a^{2}\vert \omega )(a^{0}\vert a^{4})(a^{1}\vert a^{3})+(a^{3}\vert \omega )(a^{0}\vert a^{1})(a^{2}\vert a^{4})+(a^{3}\vert \omega )(a^{0}\vert a^{2})(a^{1}\vert a^{4})+(a^{3}\vert \omega )(a^{0}\vert a^{4})(a^{1}\vert a^{2})+(a^{4}\vert \omega )(a^{0}\vert a^{1})(a^{2}\vert a^{3})+(a^{4}\vert \omega )(a^{0}\vert a^{2})(a^{1}\vert a^{3}) + (a^{4}\vert \omega )(a^{0}\vert a^{3})(a^{1}\vert a^{2})$, $\mathrm{Sym}(a^{0}\vert a^{1})(a^{2}\vert a^{3}\vert a^{4}) = (a^{0}\vert a^{1})(a^{2}\vert a^{3}\vert a^{4})+(a^{0}\vert a^{2})(a^{1}\vert a^{3}\vert a^{4})+(a^{0}\vert a^{3})(a^{1}\vert a^{2}\vert a^{4})+(a^{0}\vert a^{4})(a^{1}\vert a^{2}\vert a^{3})+(a^{1}\vert a^{2})(a^{0}\vert a^{3}\vert a^{4})+(a^{1}\vert a^{3})(a^{0}\vert a^{2}\vert a^{4})+(a^{1}\vert a^{4})(a^{0}\vert a^{2}\vert a^{3})+(a^{2}\vert a^{3})(a^{0}\vert a^{1}\vert a^{4})+(a^{2}\vert a^{4})(a^{0}\vert a^{1}\vert a^{3})+(a^{3}\vert a^{4})(a^{0}\vert a^{1}\vert a^{2})$.

$F_{0}^{(3)} = 8 {\ast} h {\ast} d {\ast} ((70 {\ast} h^{2} + 42 {\ast} h + 8) {\ast} c + 29 {\ast} h^{2} - 57 {\ast} h - 2)$, $F_{1}^{(3)} = -4 {\ast}h{\ast}d{\ast} ((14 {\ast}h + 4) {\ast}c^{2} + (-308 {\ast}h^{2} - 93 {\ast}h- 1) {\ast}c + 170 {\ast}h^{2} + 34 {\ast}h)$, $F_{2}^{(3)} = h{\ast}d{\ast} (4 {\ast}c^{3} + (-222 {\ast}h- 1) {\ast}c^{2} + (3008 {\ast}h^{2} + 102 {\ast}h) {\ast}c- 1496 {\ast}h^{2})$, $F_{0}^{(4)} = 16{\ast}h{\ast}d{\ast}((1050{\ast}h^{3}+1260{\ast}h^{2}+606{\ast}h+108){\ast}c^{2}+(3305{\ast}h^{3}-498{\ast}h^{2}-701{\ast}h+78){\ast}c-251{\ast}h^{3}+918{\ast}h^{2}-829{\ast}h-6)$, $F_{1}^{(4)} = -8{\ast}h{\ast}d{\ast}((210{\ast}h^{2}+162{\ast}h+36){\ast}c^{3}+(-4620{\ast}h^{3}-3227{\ast}h^{2}-861{\ast}h+26){\ast}c^{2}+(-5614{\ast}h^{3}+2915{\ast}h^{2}-485{\ast}h-2){\ast}c-1334{\ast}h^{3}+2622{\ast}h^{2}+92{\ast}h)$, F ₂ ⁽⁴⁾ = 2∗h∗d∗(60∗h∗c ⁴+(−3330∗h ²−523∗h−487)∗c ³+(45120∗h ³+9648∗h ²+13856∗h−2336)∗c ²+(36376∗h ³−91186∗h ²+43550∗h−2232)∗c−6760∗h ³−47796∗h ²+19756∗h−696), $F_{3}^{(4)} = 4{\ast}h{\ast}d{\ast}((42{\ast}h+36){\ast}c^{4}+(-1848{\ast}h^{2}-1279{\ast}h+513){\ast}c^{3}+(20328{\ast}h^{3}+13052{\ast}h^{2}-14654{\ast}h+2334){\ast}c^{2}+(-35836{\ast}h^{3}+98516{\ast}h^{2}-43320{\ast}h+2232){\ast}c-16700{\ast}h^{3}+43104{\ast}h^{2}-19756{\ast}h+696)$, F ₄ ⁽⁴⁾ = (1∕2)∗h∗d∗((1128∗h+199)∗c ⁴+(−46392∗h ²−3311∗h−1768)∗c ³+(497472∗h ³−19488∗h ²+73544∗h−16440)∗c ²+(351008∗h ³−726256∗h ²+326804∗h−17160)∗c−72848∗h ³−344832∗h ²+158048∗h−5568), $F_{5}^{(4)} = (-1/2){\ast}h{\ast}d{\ast}(60{\ast}c^{5}+(-2976{\ast}h+1023){\ast}c^{4}+(44184{\ast}h^{2}-41669{\ast}h+2850){\ast}c^{3}+(-164544{\ast}h^{3}+426432{\ast}h^{2}-65116{\ast}h-716){\ast}c^{2}+(-22112{\ast}h^{3}+23984{\ast}h^{2}+13092{\ast}h-1528){\ast}c+68816{\ast}h^{3}-150144{\ast}h^{2}+25024{\ast}h)$, $F_{6}^{(4)} = (1/2){\ast}h{\ast}d{\ast}(60{\ast}c^{5}+(-2640{\ast}h+1311){\ast}c^{4}+(29400{\ast}h^{2}-51901{\ast}h+6954){\ast}c^{3}+(-1920{\ast}h^{3}+530848{\ast}h^{2}-182348{\ast}h+17956){\ast}c^{2}+(-308800{\ast}h^{3}+812112{\ast}h^{2}-333468{\ast}h+16328){\ast}c-64784{\ast}h^{3}+194688{\ast}h^{2}-133024{\ast}h+5568)$, $F_{0}^{(5)} = 32{\ast}h{\ast}d{\ast}((11550{\ast}h^{4}+23100{\ast}h^{3}+20130{\ast}h^{2}+8580{\ast}h+1440){\ast}c^{2}+(76675{\ast}h^{4}+30590{\ast}h^{3}-25615{\ast}h^{2}-10898{\ast}h+1608){\ast}c+3767{\ast}h^{4}-18410{\ast}h^{3}+29929{\ast}h^{2}-16342{\ast}h-24)$, $F_{1}^{(5)} = -16{\ast}h{\ast}d{\ast}((2310{\ast}h^{3}+3366{\ast}h^{2}+1848{\ast}h+360){\ast}c^{3}+(-50820{\ast}h^{4}-64063{\ast}h^{3}-39624{\ast}h^{2}-9203{\ast}h+402){\ast}c^{2}+(-190058{\ast}h^{4}+21757{\ast}h^{3}+50420{\ast}h^{2}-8593{\ast}h-6){\ast}c+14558{\ast}h^{4}-53244{\ast}h^{3}+48082{\ast}h^{2}+348{\ast}h)$, $F_{2}^{(5)} = (4/5){\ast}h{\ast}d{\ast}((3300{\ast}h^{2}+660{\ast}h-40){\ast}c^{4}+(-183150{\ast}h^{3}-90835{\ast}h^{2}-94567{\ast}h-25578){\ast}c^{3}+(2481600{\ast}h^{4}+1334700{\ast}h^{3}+2540131{\ast}h^{2}+285789{\ast}h-163830){\ast}c^{2}+(7115560{\ast}h^{4}-13778670{\ast}h^{3}+2299334{\ast}h^{2}+2630452{\ast}h-245456){\ast}c+858872{\ast}h^{4}+1045920{\ast}h^{3}-6623912{\ast}h^{2}+2211696{\ast}h-37056)$, $F_{3}^{(5)} = (8/5){\ast}h{\ast}d{\ast}((2310{\ast}h^{2}+3300{\ast}h+920){\ast}c^{4}+(-101640{\ast}h^{3}-123925{\ast}h^{2}+2681{\ast}h+13794){\ast}c^{3}+(1118040{\ast}h^{4}+1178540{\ast}h^{3}-631298{\ast}h^{2}-179402{\ast}h+81900){\ast}c^{2}+(-228580{\ast}h^{4}+4993420{\ast}h^{3}-750692{\ast}h^{2}-1313196{\ast}h+122728){\ast}c+344284{\ast}h^{4}-2043720{\ast}h^{3}+3258596{\ast}h^{2}-1105848{\ast}h+18528)$, $F_{4}^{(5)} = (1/5){\ast}h{\ast}d{\ast}(500{\ast}c^{5}+(62040{\ast}h^{2}+7735{\ast}h+25115){\ast}c^{4}+(-2551560{\ast}h^{3}-564175{\ast}h^{2}-1452063{\ast}h+94428){\ast}c^{3}+(27360960{\ast}h^{4}+2744160{\ast}h^{3}+30534534{\ast}h^{2}-6099454{\ast}h-348380){\ast}c^{2}+(64210400{\ast}h^{4}-208744320{\ast}h^{3}+91532216{\ast}h^{2}+3799848{\ast}h-935504){\ast}c-909872{\ast}h^{4}-64093920{\ast}h^{3}+8306672{\ast}h^{2}+7635744{\ast}h-148224)$, $F_{5}^{(5)} = (-1/5){\ast}h{\ast}d{\ast}((3300{\ast}h+1500){\ast}c^{5}+(-163680{\ast}h^{2}-585{\ast}h+39235){\ast}c^{4}+(2430120{\ast}h^{3}-2017145{\ast}h^{2}-1433609{\ast}h+240084){\ast}c^{3}+(-9049920{\ast}h^{4}+31066560{\ast}h^{3}+13487402{\ast}h^{2}-7601082{\ast}h+456180){\ast}c^{2}+(-41190560{\ast}h^{4}-16962080{\ast}h^{3}+49902728{\ast}h^{2}-9779816{\ast}h-107152){\ast}c-808336{\ast}h^{4}+9987680{\ast}h^{3}-17678384{\ast}h^{2}+1778272{\ast}h+188928)$, $F_{6}^{(5)} = (1/5){\ast}h{\ast}d{\ast}((3300{\ast}h+1500){\ast}c^{5}+(-145200{\ast}h^{2}+25815{\ast}h+46595){\ast}c^{4}+(1617000{\ast}h^{3}-3008545{\ast}h^{2}-1412161{\ast}h+350436){\ast}c^{3}+(-105600{\ast}h^{4}+40494880{\ast}h^{3}+8437018{\ast}h^{2}-9036298{\ast}h+1111380){\ast}c^{2}+(-43019200{\ast}h^{4}+22985280{\ast}h^{3}+43897192{\ast}h^{2}-20285384{\ast}h+874672){\ast}c+1945936{\ast}h^{4}-6362080{\ast}h^{3}+8390384{\ast}h^{2}-7068512{\ast}h+337152)$, $F_{7}^{(5)} = (-2/5){\ast}h{\ast}d{\ast}((660{\ast}h+460){\ast}c^{5}+(-51150{\ast}h^{2}-33647{\ast}h+6897){\ast}c^{4}+(1302180{\ast}h^{3}+829156{\ast}h^{2}-451426{\ast}h+40950){\ast}c^{3}+(-10919040{\ast}h^{4}-7782640{\ast}h^{3}+9315274{\ast}h^{2}-2071698{\ast}h+61364){\ast}c^{2}+(21416272{\ast}h^{4}-59346500{\ast}h^{3}+27298188{\ast}h^{2}-1866624{\ast}h+9264){\ast}c+9686000{\ast}h^{4}-25000320{\ast}h^{3}+11458480{\ast}h^{2}-403680{\ast}h)$, $F_{8}^{(5)} = (-1/2){\ast}h{\ast}d{\ast}(100{\ast}c^{6}+(-8078{\ast}h+2861){\ast}c^{5}+(221174{\ast}h^{2}-203081{\ast}h+19684){\ast}c^{4}+(-2214880{\ast}h^{3}+4802538{\ast}h^{2}-965274{\ast}h+52252){\ast}c^{3}+(4236288{\ast}h^{4}-38346896{\ast}h^{3}+13282628{\ast}h^{2}-1695920{\ast}h+25584){\ast}c^{2}+(12825792{\ast}h^{4}-32289856{\ast}h^{3}+12276272{\ast}h^{2}-758816{\ast}h+17536){\ast}c-155904{\ast}h^{4}-1722368{\ast}h^{3}+4176000{\ast}h^{2}-215296{\ast}h)$, $F_{01423}^{(5)} = (-1/10){\ast}h{\ast}d{\ast}(500{\ast}c^{6}+(-33130{\ast}h+25625){\ast}c^{5}+(707230{\ast}h^{2}-1434751{\ast}h+485426){\ast}c^{4}+(-4128560{\ast}h^{3}+27827338{\ast}h^{2}-20020070{\ast}h+4414912){\ast}c^{3}+(-15989760{\ast}h^{4}-192127280{\ast}h^{3}+258582588{\ast}h^{2}-134080200{\ast}h+15339472){\ast}c^{2}+(136946816{\ast}h^{4}-932999600{\ast}h^{3}+1000619648{\ast}h^{2}-291521120{\ast}h+11640256){\ast}c+25836352{\ast}h^{4}-152808960{\ast}h^{3}+252283328{\ast}h^{2}-102259584{\ast}h+4475904)$, $F_{01324}^{(5)} = (1/10){\ast}h{\ast}d{\ast}(500{\ast}c^{6}+(-35770{\ast}h+23785){\ast}c^{5}+(911830{\ast}h^{2}-1300163{\ast}h+457838){\ast}c^{4}+(-9337280{\ast}h^{3}+24510714{\ast}h^{2}-18214366{\ast}h+4251112){\ast}c^{3}+(27686400{\ast}h^{4}-160996720{\ast}h^{3}+221321492{\ast}h^{2}-125793408{\ast}h+15094016){\ast}c^{2}+(51281728{\ast}h^{4}-695613600{\ast}h^{3}+891426896{\ast}h^{2}-284054624{\ast}h+11603200){\ast}c-12907648{\ast}h^{4}-52807680{\ast}h^{3}+206449408{\ast}h^{2}-100644864{\ast}h+4475904)$, $F_{12034}^{(5)} = (1/10){\ast}h{\ast}d{\ast}(100{\ast}c^{6}+(-3150{\ast}h+5575){\ast}c^{5}+(15650{\ast}h^{2}-148721{\ast}h+119806){\ast}c^{4}+(-550800{\ast}h^{3}-1490922{\ast}h^{2}-4041146{\ast}h+949728){\ast}c^{3}+(14745600{\ast}h^{4}+53833840{\ast}h^{3}+29411876{\ast}h^{2}-25511768{\ast}h+3284592){\ast}c^{2}+(-167754624{\ast}h^{4}+73524400{\ast}h^{3}+137639360{\ast}h^{2}-61086944{\ast}h+2624448){\ast}c+3115968{\ast}h^{4}-16839680{\ast}h^{3}+31808832{\ast}h^{2}-21690496{\ast}h+1007616)$, $F_{01234}^{(5)} = h{\ast}d{\ast}(50{\ast}c^{6}+(-5304{\ast}h+1238){\ast}c^{5}+(204604{\ast}h^{2}-99615{\ast}h+13827){\ast}c^{4}+(-3383208{\ast}h^{3}+2721160{\ast}h^{2}-892294{\ast}h+59250){\ast}c^{3}+(20120832{\ast}h^{4}-26868960{\ast}h^{3}+17636364{\ast}h^{2}-3522876{\ast}h-65568){\ast}c^{2}+(41237472{\ast}h^{4}-107859720{\ast}h^{3}+46229896{\ast}h^{2}-423632{\ast}h-256192){\ast}c-772320{\ast}h^{4}-27659904{\ast}h^{3}+9574560{\ast}h^{2}+1892928{\ast}h-31488)$, $F_{02134}^{(5)} = (-1/10){\ast}h{\ast}d{\ast}(100{\ast}c^{6}+(150{\ast}h+7215){\ast}c^{5}+(-237790{\ast}h^{2}-280007{\ast}h+130072){\ast}c^{4}+(5835360{\ast}h^{3}+1550206{\ast}h^{2}-5491014{\ast}h+812348){\ast}c^{3}+(-38223360{\ast}h^{4}+30779440{\ast}h^{3}+63018188{\ast}h^{2}-27827312{\ast}h+1723824){\ast}c^{2}+(-139274688{\ast}h^{4}-126848480{\ast}h^{3}+208934224{\ast}h^{2}-38241856{\ast}h-180160){\ast}c+69888{\ast}h^{4}+19482880{\ast}h^{3}-41267328{\ast}h^{2}+3697664{\ast}h+692736)$, $F_{03214}^{(5)} = F_{04213}^{(5)} = (1/5){\ast}h{\ast}d{\ast}((1650{\ast}h+820){\ast}c^{5}+(-126720{\ast}h^{2}-65643{\ast}h+5133){\ast}c^{4}+(3193080{\ast}h^{3}+1520564{\ast}h^{2}-724934{\ast}h-68690){\ast}c^{3}+(-26484480{\ast}h^{4}-11527200{\ast}h^{3}+16803156{\ast}h^{2}-1157772{\ast}h-780384){\ast}c^{2}+(14239968{\ast}h^{4}-100186440{\ast}h^{3}+35647432{\ast}h^{2}+11422544{\ast}h-1402304){\ast}c-1523040{\ast}h^{4}+18161280{\ast}h^{3}-36538080{\ast}h^{2}+12694080{\ast}h-157440)$, $F_{02413}^{(5)} = F_{03412}^{(5)} = (-1/5){\ast}h{\ast}d{\ast}((3630{\ast}h+5660){\ast}c^{5}+(-199320{\ast}h^{2}-209673{\ast}h+193503){\ast}c^{4}+(3472920{\ast}h^{3}+1907324{\ast}h^{2}-7596850{\ast}h+2076826){\ast}c^{3}+(-18585600{\ast}h^{4}-196400{\ast}h^{3}+96084724{\ast}h^{2}-62800300{\ast}h+7605776){\ast}c^{2}+(36408928{\ast}h^{4}-385775160{\ast}h^{3}+469619144{\ast}h^{2}-143863520{\ast}h+5776288){\ast}c+13307936{\ast}h^{4}-72098560{\ast}h^{3}+115701664{\ast}h^{2}-50591552{\ast}h+2237952)$, $F_{02314}^{(5)} = F_{04312}^{(5)} = (1/5){\ast}h{\ast}d{\ast}((2310{\ast}h+4740){\ast}c^{5}+(-97020{\ast}h^{2}-142379{\ast}h+179709){\ast}c^{4}+(868560{\ast}h^{3}+249012{\ast}h^{2}-6693998{\ast}h+1994926){\ast}c^{3}+(3252480{\ast}h^{4}+15368880{\ast}h^{3}+77454176{\ast}h^{2}-58656904{\ast}h+7483048){\ast}c^{2}+(-6423616{\ast}h^{4}-267082160{\ast}h^{3}+415022768{\ast}h^{2}-140130272{\ast}h+5757760){\ast}c-6064064{\ast}h^{4}-22097920{\ast}h^{3}+92784704{\ast}h^{2}-49784192{\ast}h+2237952)$, $F_{04123}^{(5)} = F_{03124}^{(5)} = (-2/5){\ast}h{\ast}d{\ast}(150{\ast}c^{6}+(-10060{\ast}h+5380){\ast}c^{5}+(217020{\ast}h^{2}-323853{\ast}h+57123){\ast}c^{4}+(-1309760{\ast}h^{3}+6390724{\ast}h^{2}-2579346{\ast}h+268402){\ast}c^{3}+(-4260480{\ast}h^{4}-40238760{\ast}h^{3}+32357832{\ast}h^{2}-9076728{\ast}h+462936){\ast}c^{2}+(-18786432{\ast}h^{4}-72074440{\ast}h^{3}+67578896{\ast}h^{2}-10508984{\ast}h-23120){\ast}c-177408{\ast}h^{4}+2717760{\ast}h^{3}-5096832{\ast}h^{2}+655296{\ast}h+173184)$, $F_{14023}^{(5)} = F_{13024}^{(5)} = (1/5){\ast}h{\ast}d{\ast}(300{\ast}c^{6}+(-21770{\ast}h+9940){\ast}c^{5}+(560760{\ast}h^{2}-582063{\ast}h+109113){\ast}c^{4}+(-5812600{\ast}h^{3}+11260884{\ast}h^{2}-4433758{\ast}h+605494){\ast}c^{3}+(17963520{\ast}h^{4}-68950320{\ast}h^{3}+47912508{\ast}h^{2}-16995684{\ast}h+1706256){\ast}c^{2}+(-51812832{\ast}h^{4}-43962440{\ast}h^{3}+99510360{\ast}h^{2}-32440512{\ast}h+1356064){\ast}c+1168224{\ast}h^{4}-12725760{\ast}h^{3}+26344416{\ast}h^{2}-11383488{\ast}h+503808)$, $E_{3}^{(3)} = 8 {\ast} h {\ast} d {\ast} ((70 {\ast} h^{2} + 42 {\ast} h + 8) {\ast} c + 29 {\ast} h^{2} - 57 {\ast} h - 2)$, $E_{2}^{(3)} = -12{\ast}h{\ast}d{\ast}((14{\ast}h+4){\ast}c^{2} +(-308{\ast}h^{2} -93{\ast}h-1){\ast}c+170{\ast}h^{2} +34{\ast}h)$, $E_{1}^{(3)} = 2 {\ast}h{\ast}d{\ast} (4 {\ast}c^{3} + (-222 {\ast}h- 1) {\ast}c^{2} + (3008 {\ast}h^{2} + 102 {\ast}h) {\ast}c- 1496 {\ast}h^{2})$, $E_{4}^{(4)} = 16{\ast}h{\ast}d{\ast}((1050{\ast}h^{3}+1260{\ast}h^{2}+606{\ast}h+108){\ast}c^{2}+(3305{\ast}h^{3}-498{\ast}h^{2}-701{\ast}h+78){\ast}c-251{\ast}h^{3}+918{\ast}h^{2}-829{\ast}h-6)$, E ₃ ⁽⁴⁾ = −48∗h∗d∗((210∗h ²+162∗h+36)∗c ³+(−4620∗h ³−3227∗h ²−861∗h+26)∗c ²+(−5614∗h ³+2915∗h ²−485∗h−2)∗c−1334∗h ³+2622∗h ²+92∗h), $E_{2}^{(4)} = 4{\ast}h{\ast}d{\ast}((366{\ast}h+108){\ast}c^{4}+(-18864{\ast}h^{2}-5929{\ast}h-409){\ast}c^{3}+(241464{\ast}h^{3}+77748{\ast}h^{2}+11462{\ast}h-2342){\ast}c^{2}+(37996{\ast}h^{3}-69196{\ast}h^{2}+44240{\ast}h-2232){\ast}c-77140{\ast}h^{3}-61872{\ast}h^{2}+19756{\ast}h-696)$, $E_{1}^{(4)} = 2{\ast}h{\ast}d{\ast}((1464{\ast}h+487){\ast}c^{4}+(-61176{\ast}h^{2}-13543{\ast}h+2336){\ast}c^{3}+(660096{\ast}h^{3}+84928{\ast}h^{2}-43688{\ast}h+2232){\ast}c^{2}+(64320{\ast}h^{3}+61872{\ast}h^{2}-19756{\ast}h+696){\ast}c-206448{\ast}h^{3})$, $E_{5}^{(5)} = 32{\ast}h{\ast}d{\ast}((11550{\ast}h^{4}+23100{\ast}h^{3}+20130{\ast}h^{2}+8580{\ast}h+1440){\ast}c^{2}+(76675{\ast}h^{4}+30590{\ast}h^{3}-25615{\ast}h^{2}-10898{\ast}h+1608){\ast}c+3767{\ast}h^{4}-18410{\ast}h^{3}+29929{\ast}h^{2}-16342{\ast}h-24)$, $E_{4}^{(5)} = -160{\ast}h{\ast}d{\ast}((2310{\ast}h^{3}+3366{\ast}h^{2}+1848{\ast}h+360){\ast}c^{3}+(-50820{\ast}h^{4}-64063{\ast}h^{3}-39624{\ast}h^{2}-9203{\ast}h+402){\ast}c^{2}+(-190058{\ast}h^{4}+21757{\ast}h^{3}+50420{\ast}h^{2}-8593{\ast}h-6){\ast}c+14558{\ast}h^{4}-53244{\ast}h^{3}+48082{\ast}h^{2}+348{\ast}h)$, $E_{3}^{(5)} = 8{\ast}h{\ast}d{\ast}((13530{\ast}h^{2}+11220{\ast}h+2680){\ast}c^{4}+(-671220{\ast}h^{3}-553445{\ast}h^{2}-181091{\ast}h-9774){\ast}c^{3}+(8317320{\ast}h^{4}+6205020{\ast}h^{3}+3186368{\ast}h^{2}+33372{\ast}h-81960){\ast}c^{2}+(13545380{\ast}h^{4}-12577080{\ast}h^{3}+2346592{\ast}h^{2}+1321316{\ast}h-122728){\ast}c+2750596{\ast}h^{4}-4039320{\ast}h^{3}-3472036{\ast}h^{2}+1105848{\ast}h-18528)$, E ₂ ⁽⁵⁾ = −4∗h∗d∗((1320∗h+420)∗c ⁵+(−182820∗h ²−101429∗h−18681)∗c ⁴+(5969040∗h ³+3213887∗h ²+527763∗h−122880)∗c ³+(−58143360∗h ⁴−27737760∗h ³−6853602∗h ²+3391274∗h−184092)∗c ²+(−19549216∗h ⁴+50103960∗h ³−30930304∗h ²+2972472∗h−27792)∗c+17527600∗h ⁴+30443040∗h ³−11458480∗h ²+403680∗h), $E_{1}^{(5)} = -2{\ast}h{\ast}d{\ast}(100{\ast}c^{6}+(1470{\ast}h+6495){\ast}c^{5}+(-501790{\ast}h^{2}-424803{\ast}h+40956){\ast}c^{4}+(15693120{\ast}h^{3}+8719374{\ast}h^{2}-2073438{\ast}h+61364){\ast}c^{3}+(-141373440{\ast}h^{4}-54143280{\ast}h^{3}+27458268{\ast}h^{2}-1866624{\ast}h+9264){\ast}c^{2}+(-12282432{\ast}h^{4}-30443040{\ast}h^{3}+11458480{\ast}h^{2}-403680{\ast}h){\ast}c+47895936{\ast}h^{4})$.