# Properties

 Label 3.67.b.b Level 3 Weight 67 Character orbit 3.b Analytic conductor 82.760 Analytic rank 0 Dimension 20 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3$$ Weight: $$k$$ = $$67$$ Character orbit: $$[\chi]$$ = 3.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$82.7604085389$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: multiple of $$2^{216}\cdot 3^{291}\cdot 5^{20}\cdot 7^{10}\cdot 11^{6}\cdot 13^{4}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+\beta_{1} q^{2}$$ $$+(243548779847726 + 15513 \beta_{1} - \beta_{2}) q^{3}$$ $$+(-43723635666549374440 + 141 \beta_{1} - 1519 \beta_{2} + \beta_{3}) q^{4}$$ $$+(-98651 + 333501149933 \beta_{1} + 493259 \beta_{2} + 3 \beta_{3} + \beta_{4}) q^{5}$$ $$+(-$$$$18\!\cdots\!70$$$$- 154836520809205 \beta_{1} - 23575093 \beta_{2} - 44166 \beta_{3} + 45 \beta_{4} - \beta_{5}) q^{6}$$ $$+($$$$63\!\cdots\!16$$$$+ 807734091 \beta_{1} - 8693458123 \beta_{2} + 4109226 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6}) q^{7}$$ $$+(3272193665213 - 42419403620205012804 \beta_{1} - 16361058157166 \beta_{2} - 90503341 \beta_{3} + 671787 \beta_{4} + 227 \beta_{5} + 7 \beta_{7} + \beta_{8}) q^{8}$$ $$+($$$$75\!\cdots\!72$$$$+$$$$56\!\cdots\!53$$$$\beta_{1} - 204864643581341 \beta_{2} - 16029368004 \beta_{3} - 10845989 \beta_{4} - 19747 \beta_{5} + 1112 \beta_{6} - 414 \beta_{7} - 5 \beta_{8} + \beta_{9}) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+\beta_{1} q^{2}$$ $$+(243548779847726 + 15513 \beta_{1} - \beta_{2}) q^{3}$$ $$+(-43723635666549374440 + 141 \beta_{1} - 1519 \beta_{2} + \beta_{3}) q^{4}$$ $$+(-98651 + 333501149933 \beta_{1} + 493259 \beta_{2} + 3 \beta_{3} + \beta_{4}) q^{5}$$ $$+(-$$$$18\!\cdots\!70$$$$- 154836520809205 \beta_{1} - 23575093 \beta_{2} - 44166 \beta_{3} + 45 \beta_{4} - \beta_{5}) q^{6}$$ $$+($$$$63\!\cdots\!16$$$$+ 807734091 \beta_{1} - 8693458123 \beta_{2} + 4109226 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6}) q^{7}$$ $$+(3272193665213 - 42419403620205012804 \beta_{1} - 16361058157166 \beta_{2} - 90503341 \beta_{3} + 671787 \beta_{4} + 227 \beta_{5} + 7 \beta_{7} + \beta_{8}) q^{8}$$ $$+($$$$75\!\cdots\!72$$$$+$$$$56\!\cdots\!53$$$$\beta_{1} - 204864643581341 \beta_{2} - 16029368004 \beta_{3} - 10845989 \beta_{4} - 19747 \beta_{5} + 1112 \beta_{6} - 414 \beta_{7} - 5 \beta_{8} + \beta_{9}) q^{9}$$ $$+(-$$$$39\!\cdots\!80$$$$- 2006354176353009 \beta_{1} + 21545263826215911 \beta_{2} - 712860668503 \beta_{3} - 21061229 \beta_{4} - 2227435 \beta_{5} - 42610 \beta_{6} + 6414 \beta_{7} + 5 \beta_{8} - \beta_{10}) q^{10}$$ $$+(-24769660378890561 +$$$$81\!\cdots\!44$$$$\beta_{1} + 123848982841784823 \beta_{2} + 685201679889 \beta_{3} - 4484691838 \beta_{4} + 115140705 \beta_{5} - 29842 \beta_{6} + 47894 \beta_{7} - 2871 \beta_{8} - 78 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14}) q^{11}$$ $$+($$$$36\!\cdots\!67$$$$+$$$$24\!\cdots\!32$$$$\beta_{1} + 44281020554661386717 \beta_{2} + 534479422112123 \beta_{3} - 676198095841 \beta_{4} - 216503077 \beta_{5} - 24701951 \beta_{6} + 1835810 \beta_{7} - 91989 \beta_{8} + 3394 \beta_{9} - 61 \beta_{10} - 18 \beta_{11} + 2 \beta_{12} - 3 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17}) q^{12}$$ $$+($$$$90\!\cdots\!05$$$$+ 168306987341434474 \beta_{1} - 1809557276040313878 \beta_{2} + 487056592351563 \beta_{3} + 910533351 \beta_{4} - 1056867305 \beta_{5} - 146206944 \beta_{6} - 956917 \beta_{7} - 22282 \beta_{8} + 51454 \beta_{9} + 91 \beta_{10} - 13 \beta_{11} - 26 \beta_{12} - 13 \beta_{13} - 13 \beta_{15}) q^{13}$$ $$+($$$$11\!\cdots\!66$$$$+$$$$33\!\cdots\!34$$$$\beta_{1} -$$$$56\!\cdots\!40$$$$\beta_{2} - 3101175656259103 \beta_{3} + 177737010422686 \beta_{4} - 389149947578 \beta_{5} + 96713155 \beta_{6} - 61850541 \beta_{7} - 10595111 \beta_{8} - 224556 \beta_{9} + 2917 \beta_{10} + 2871 \beta_{11} + 15 \beta_{12} + 9 \beta_{13} + 312 \beta_{14} - 25 \beta_{15} - 4 \beta_{16} + 28 \beta_{17} - \beta_{18}) q^{14}$$ $$+($$$$14\!\cdots\!73$$$$+$$$$57\!\cdots\!70$$$$\beta_{1} +$$$$45\!\cdots\!55$$$$\beta_{2} + 1191470331005851167 \beta_{3} - 209192030057892 \beta_{4} - 537684921841 \beta_{5} - 5373055708 \beta_{6} - 1783497101 \beta_{7} - 104208626 \beta_{8} - 1125947 \beta_{9} + 4627 \beta_{10} + 6655 \beta_{11} - 2737 \beta_{12} + 1005 \beta_{13} - 91 \beta_{14} - 44 \beta_{15} - 685 \beta_{16} - 203 \beta_{17} - \beta_{18} - \beta_{19}) q^{15}$$ $$+($$$$17\!\cdots\!35$$$$+$$$$53\!\cdots\!52$$$$\beta_{1} -$$$$57\!\cdots\!76$$$$\beta_{2} - 59837743460457257784 \beta_{3} + 144494037007035 \beta_{4} - 27337238790086 \beta_{5} + 162598667649 \beta_{6} - 84058466325 \beta_{7} + 81866204 \beta_{8} + 2058155 \beta_{9} - 1385528 \beta_{10} - 227556 \beta_{11} + 26976 \beta_{12} - 17257 \beta_{13} + 29517 \beta_{15} - 13575 \beta_{16}) q^{16}$$ $$+(-$$$$82\!\cdots\!03$$$$+$$$$25\!\cdots\!10$$$$\beta_{1} +$$$$41\!\cdots\!37$$$$\beta_{2} + 2272966005922283479 \beta_{3} - 53267819756940843 \beta_{4} + 51171855945150 \beta_{5} - 11757628214 \beta_{6} - 138617535513 \beta_{7} - 4118387223 \beta_{8} + 93532347 \beta_{9} - 376024 \beta_{10} - 354505 \beta_{11} - 10722 \beta_{12} - 9700 \beta_{13} - 73624 \beta_{14} + 8547 \beta_{15} - 1467 \beta_{16} - 24562 \beta_{17} + 154 \beta_{18} - 54 \beta_{19}) q^{17}$$ $$+(-$$$$66\!\cdots\!27$$$$+$$$$18\!\cdots\!10$$$$\beta_{1} +$$$$14\!\cdots\!00$$$$\beta_{2} +$$$$10\!\cdots\!03$$$$\beta_{3} + 221181866557737097 \beta_{4} + 113870334956445 \beta_{5} + 1752875203963 \beta_{6} + 775074624591 \beta_{7} - 46623522768 \beta_{8} - 131877765 \beta_{9} + 47095812 \beta_{10} + 11499494 \beta_{11} + 562912 \beta_{12} + 50053 \beta_{13} + 45384 \beta_{14} - 711400 \beta_{15} + 714382 \beta_{16} - 48804 \beta_{17} - 1893 \beta_{18} - 192 \beta_{19}) q^{18}$$ $$+(-$$$$10\!\cdots\!36$$$$+$$$$22\!\cdots\!75$$$$\beta_{1} -$$$$24\!\cdots\!76$$$$\beta_{2} -$$$$12\!\cdots\!25$$$$\beta_{3} + 27044631308377545 \beta_{4} + 3182314905096243 \beta_{5} + 28926411853036 \beta_{6} - 7859576466609 \beta_{7} - 10975544351 \beta_{8} + 8582057348 \beta_{9} + 141401103 \beta_{10} + 19863444 \beta_{11} + 1447261 \beta_{12} - 4655558 \beta_{13} - 3905707 \beta_{15} + 4253124 \beta_{16}) q^{19}$$ $$+(-$$$$61\!\cdots\!27$$$$+$$$$45\!\cdots\!98$$$$\beta_{1} +$$$$30\!\cdots\!83$$$$\beta_{2} +$$$$16\!\cdots\!42$$$$\beta_{3} - 23988241282028380966 \beta_{4} + 40339083785237983 \beta_{5} - 9084273075325 \beta_{6} - 98703458731981 \beta_{7} - 955146182871 \beta_{8} + 51372893859 \beta_{9} - 282268265 \beta_{10} - 270082284 \beta_{11} - 6907347 \beta_{12} - 6646019 \beta_{13} - 20453922 \beta_{14} + 4162766 \beta_{15} - 1527457 \beta_{16} - 16366006 \beta_{17} + 74368 \beta_{18} - 6912 \beta_{19}) q^{20}$$ $$+($$$$42\!\cdots\!02$$$$+$$$$15\!\cdots\!73$$$$\beta_{1} -$$$$62\!\cdots\!34$$$$\beta_{2} -$$$$17\!\cdots\!94$$$$\beta_{3} + 19438807904075676293 \beta_{4} + 20389866953639864 \beta_{5} + 2237871417454348 \beta_{6} - 988007336580931 \beta_{7} - 6961085268831 \beta_{8} - 346939732130 \beta_{9} - 23881348 \beta_{10} - 357948321 \beta_{11} + 10406435 \beta_{12} - 1418493 \beta_{13} - 439422240 \beta_{14} + 114613126 \beta_{15} - 29955634 \beta_{16} - 36965614 \beta_{17} - 257562 \beta_{18} + 10710 \beta_{19}) q^{21}$$ $$+(-$$$$96\!\cdots\!79$$$$+$$$$12\!\cdots\!78$$$$\beta_{1} -$$$$13\!\cdots\!86$$$$\beta_{2} +$$$$18\!\cdots\!15$$$$\beta_{3} + 13505675640979296233 \beta_{4} + 937188583246912087 \beta_{5} + 7317100870984344 \beta_{6} - 4527124844632888 \beta_{7} - 1580498234247 \beta_{8} - 495323279011 \beta_{9} + 7302193857 \beta_{10} + 5109872495 \beta_{11} - 2646055426 \beta_{12} - 122565280 \beta_{13} - 752214578 \beta_{15} + 287360604 \beta_{16}) q^{22}$$ $$+(-$$$$86\!\cdots\!22$$$$+$$$$39\!\cdots\!46$$$$\beta_{1} +$$$$43\!\cdots\!49$$$$\beta_{2} +$$$$23\!\cdots\!46$$$$\beta_{3} -$$$$59\!\cdots\!49$$$$\beta_{4} + 696744462125720731 \beta_{5} + 8499378721672 \beta_{6} - 17534752714713524 \beta_{7} - 156975098366779 \beta_{8} + 6279328349154 \beta_{9} + 31078113 \beta_{10} + 1398970352 \beta_{11} - 890693667 \beta_{12} - 915577136 \beta_{13} + 10923005600 \beta_{14} + 370855595 \beta_{15} - 275832862 \beta_{16} - 2193895018 \beta_{17} - 3519422 \beta_{18} + 1006722 \beta_{19}) q^{23}$$ $$+(-$$$$42\!\cdots\!16$$$$+$$$$39\!\cdots\!52$$$$\beta_{1} -$$$$31\!\cdots\!02$$$$\beta_{2} +$$$$10\!\cdots\!43$$$$\beta_{3} -$$$$74\!\cdots\!22$$$$\beta_{4} + 40697612525129294877 \beta_{5} + 99640010197410433 \beta_{6} - 52248700680706066 \beta_{7} - 359643966549163 \beta_{8} + 5176689414059 \beta_{9} + 1357786141460 \beta_{10} - 114231981172 \beta_{11} - 17630357444 \beta_{12} - 1489323921 \beta_{13} - 106033951520 \beta_{14} - 7215855823 \beta_{15} + 6677959609 \beta_{16} - 4948122208 \beta_{17} + 24988096 \beta_{18} + 213760 \beta_{19}) q^{24}$$ $$+(-$$$$15\!\cdots\!40$$$$+$$$$19\!\cdots\!90$$$$\beta_{1} -$$$$21\!\cdots\!14$$$$\beta_{2} -$$$$37\!\cdots\!07$$$$\beta_{3} +$$$$27\!\cdots\!68$$$$\beta_{4} + 9931640817449648345 \beta_{5} + 267676016813301518 \beta_{6} - 100758455768033375 \beta_{7} + 38237234491738 \beta_{8} - 102062127425331 \beta_{9} - 5190729384525 \beta_{10} + 123555354403 \beta_{11} + 18677248831 \beta_{12} - 12361445862 \beta_{13} + 13164679142 \beta_{15} + 23682888249 \beta_{16}) q^{25}$$ $$+(-$$$$24\!\cdots\!54$$$$+$$$$86\!\cdots\!34$$$$\beta_{1} +$$$$12\!\cdots\!20$$$$\beta_{2} +$$$$66\!\cdots\!72$$$$\beta_{3} -$$$$16\!\cdots\!34$$$$\beta_{4} -$$$$18\!\cdots\!22$$$$\beta_{5} + 48141835039897638 \beta_{6} - 213749193099444522 \beta_{7} - 8823796161416340 \beta_{8} + 562068909719070 \beta_{9} + 1503903671868 \beta_{10} + 1580452796000 \beta_{11} - 7868469882 \beta_{12} + 7272229770 \beta_{13} + 1142425308752 \beta_{14} + 55309017062 \beta_{15} + 19566488552 \beta_{16} + 970071544 \beta_{17} - 15637258 \beta_{18} - 54587520 \beta_{19}) q^{26}$$ $$+($$$$27\!\cdots\!49$$$$+$$$$16\!\cdots\!77$$$$\beta_{1} -$$$$52\!\cdots\!47$$$$\beta_{2} -$$$$28\!\cdots\!15$$$$\beta_{3} +$$$$21\!\cdots\!09$$$$\beta_{4} -$$$$51\!\cdots\!68$$$$\beta_{5} - 1476677729408076276 \beta_{6} + 1302654428383585518 \beta_{7} + 9814650205801446 \beta_{8} + 485284007792700 \beta_{9} + 87788122632504 \beta_{10} - 764592440799 \beta_{11} + 367495890771 \beta_{12} - 98242695312 \beta_{13} - 1778382274383 \beta_{14} + 447710911305 \beta_{15} - 100536968922 \beta_{16} + 156262477518 \beta_{17} - 876918294 \beta_{18} - 36503766 \beta_{19}) q^{27}$$ $$+($$$$72\!\cdots\!85$$$$-$$$$44\!\cdots\!22$$$$\beta_{1} +$$$$48\!\cdots\!87$$$$\beta_{2} +$$$$16\!\cdots\!92$$$$\beta_{3} -$$$$51\!\cdots\!72$$$$\beta_{4} -$$$$24\!\cdots\!01$$$$\beta_{5} - 65536019662454142629 \beta_{6} + 18400113042966923233 \beta_{7} + 5893954219022009 \beta_{8} - 5000185140655103 \beta_{9} - 378394054518613 \beta_{10} - 4685479887872 \beta_{11} + 1734873544607 \beta_{12} + 1179135840603 \beta_{13} + 1876365803536 \beta_{15} - 723854031789 \beta_{16}) q^{28}$$ $$+($$$$61\!\cdots\!57$$$$+$$$$42\!\cdots\!63$$$$\beta_{1} -$$$$30\!\cdots\!71$$$$\beta_{2} -$$$$16\!\cdots\!09$$$$\beta_{3} +$$$$59\!\cdots\!95$$$$\beta_{4} -$$$$12\!\cdots\!42$$$$\beta_{5} + 121666834561016212 \beta_{6} + 11945253476111186986 \beta_{7} - 449980534057236588 \beta_{8} + 30705636196064898 \beta_{9} + 580396076338 \beta_{10} + 4895998772150 \beta_{11} + 719866807878 \beta_{12} + 2040348825720 \beta_{13} - 9234470847856 \beta_{14} + 4080840349252 \beta_{15} + 2076982725862 \beta_{16} + 3494677340504 \beta_{17} + 5542446472 \beta_{18} + 1794069000 \beta_{19}) q^{29}$$ $$+(-$$$$67\!\cdots\!34$$$$-$$$$70\!\cdots\!12$$$$\beta_{1} +$$$$29\!\cdots\!88$$$$\beta_{2} +$$$$15\!\cdots\!21$$$$\beta_{3} -$$$$25\!\cdots\!14$$$$\beta_{4} -$$$$96\!\cdots\!46$$$$\beta_{5} -$$$$11\!\cdots\!69$$$$\beta_{6} + 54033481224259953607 \beta_{7} + 2875208016481719003 \beta_{8} + 10101868009202120 \beta_{9} + 1715894895749635 \beta_{10} + 150440720387079 \beta_{11} + 2899847729401 \beta_{12} + 12554926265769 \beta_{13} + 100617707580024 \beta_{14} + 15455817910817 \beta_{15} + 6340391089552 \beta_{16} + 1292468245852 \beta_{17} + 15355393119 \beta_{18} + 1657384704 \beta_{19}) q^{30}$$ $$+($$$$30\!\cdots\!10$$$$+$$$$73\!\cdots\!75$$$$\beta_{1} -$$$$78\!\cdots\!64$$$$\beta_{2} -$$$$21\!\cdots\!14$$$$\beta_{3} +$$$$11\!\cdots\!94$$$$\beta_{4} +$$$$22\!\cdots\!79$$$$\beta_{5} - 90374444641799555087 \beta_{6} - 25793630245299129516 \beta_{7} + 49693062020770031 \beta_{8} - 229070481196549496 \beta_{9} + 437251064697033 \beta_{10} + 186711635265300 \beta_{11} - 43300939660921 \beta_{12} - 20742683333014 \beta_{13} + 32825735136079 \beta_{15} + 26712084480864 \beta_{16}) q^{31}$$ $$+(-$$$$58\!\cdots\!84$$$$+$$$$29\!\cdots\!08$$$$\beta_{1} +$$$$29\!\cdots\!72$$$$\beta_{2} +$$$$16\!\cdots\!84$$$$\beta_{3} -$$$$78\!\cdots\!80$$$$\beta_{4} +$$$$53\!\cdots\!16$$$$\beta_{5} - 7565690328956741408 \beta_{6} -$$$$11\!\cdots\!68$$$$\beta_{7} - 59900743195816302480 \beta_{8} + 1022882632440221664 \beta_{9} - 481859934597664 \beta_{10} - 284903221474432 \beta_{11} - 52705914990816 \beta_{12} - 26555106345184 \beta_{13} - 806727011023936 \beta_{14} + 115585109859648 \beta_{15} + 23252169322464 \beta_{16} - 93242037165760 \beta_{17} - 216768355712 \beta_{18} - 41204602368 \beta_{19}) q^{32}$$ $$+($$$$38\!\cdots\!20$$$$-$$$$35\!\cdots\!47$$$$\beta_{1} -$$$$17\!\cdots\!44$$$$\beta_{2} -$$$$35\!\cdots\!35$$$$\beta_{3} -$$$$10\!\cdots\!66$$$$\beta_{4} +$$$$43\!\cdots\!33$$$$\beta_{5} -$$$$47\!\cdots\!22$$$$\beta_{6} -$$$$96\!\cdots\!53$$$$\beta_{7} + 92593382109836726144 \beta_{8} + 808231311904196486 \beta_{9} - 53419517891892424 \beta_{10} + 393453796633061 \beta_{11} - 326818675494404 \beta_{12} - 278204770840398 \beta_{13} - 218754111114872 \beta_{14} + 144236512488095 \beta_{15} + 117522174876481 \beta_{16} - 121905928776550 \beta_{17} - 78046105634 \beta_{18} - 46270605266 \beta_{19}) q^{33}$$ $$+(-$$$$30\!\cdots\!96$$$$+$$$$70\!\cdots\!96$$$$\beta_{1} -$$$$75\!\cdots\!18$$$$\beta_{2} +$$$$68\!\cdots\!96$$$$\beta_{3} +$$$$68\!\cdots\!62$$$$\beta_{4} +$$$$13\!\cdots\!98$$$$\beta_{5} +$$$$23\!\cdots\!64$$$$\beta_{6} -$$$$15\!\cdots\!00$$$$\beta_{7} - 2037296753024088378 \beta_{8} - 2371754963378464784 \beta_{9} + 181383093791780274 \beta_{10} + 7108370736424204 \beta_{11} - 236772159255402 \beta_{12} + 356438552128928 \beta_{13} - 25729765626138 \beta_{15} + 383399006202444 \beta_{16}) q^{34}$$ $$+($$$$15\!\cdots\!09$$$$+$$$$22\!\cdots\!61$$$$\beta_{1} -$$$$79\!\cdots\!86$$$$\beta_{2} -$$$$43\!\cdots\!68$$$$\beta_{3} -$$$$42\!\cdots\!36$$$$\beta_{4} -$$$$20\!\cdots\!83$$$$\beta_{5} +$$$$42\!\cdots\!90$$$$\beta_{6} +$$$$15\!\cdots\!81$$$$\beta_{7} -$$$$55\!\cdots\!69$$$$\beta_{8} - 4297672365199886754 \beta_{9} + 15484512503170405 \beta_{10} + 14404370497801199 \beta_{11} + 375997738698462 \beta_{12} + 270458143511904 \beta_{13} + 16559246384968367 \beta_{14} - 561215207596046 \beta_{15} - 52239882653108 \beta_{16} + 773715794484836 \beta_{17} + 4885545373612 \beta_{18} + 705065200812 \beta_{19}) q^{35}$$ $$+(-$$$$15\!\cdots\!63$$$$-$$$$99\!\cdots\!57$$$$\beta_{1} -$$$$43\!\cdots\!68$$$$\beta_{2} +$$$$43\!\cdots\!63$$$$\beta_{3} +$$$$37\!\cdots\!54$$$$\beta_{4} -$$$$16\!\cdots\!49$$$$\beta_{5} -$$$$16\!\cdots\!85$$$$\beta_{6} +$$$$52\!\cdots\!15$$$$\beta_{7} +$$$$20\!\cdots\!89$$$$\beta_{8} - 14885539198748789237 \beta_{9} - 222294880441696869 \beta_{10} + 14077789995463596 \beta_{11} + 3950896511557197 \beta_{12} + 4213898784405837 \beta_{13} - 28187902605836142 \beta_{14} - 3406239189593766 \beta_{15} - 2240276838825861 \beta_{16} + 2312711848419462 \beta_{17} - 2951491215744 \beta_{18} + 929160495360 \beta_{19}) q^{36}$$ $$+($$$$13\!\cdots\!55$$$$-$$$$10\!\cdots\!50$$$$\beta_{1} +$$$$10\!\cdots\!58$$$$\beta_{2} -$$$$58\!\cdots\!83$$$$\beta_{3} -$$$$10\!\cdots\!69$$$$\beta_{4} -$$$$74\!\cdots\!51$$$$\beta_{5} +$$$$26\!\cdots\!48$$$$\beta_{6} +$$$$33\!\cdots\!65$$$$\beta_{7} - 24063364435328543586 \beta_{8} + 93381905261382060112 \beta_{9} - 1502896701506904547 \beta_{10} - 36885515415962599 \beta_{11} + 16446456371904672 \beta_{12} - 3420748957243449 \beta_{13} - 2495248454035269 \beta_{15} - 15535116457689030 \beta_{16}) q^{37}$$ $$+($$$$70\!\cdots\!51$$$$-$$$$31\!\cdots\!20$$$$\beta_{1} -$$$$35\!\cdots\!18$$$$\beta_{2} -$$$$19\!\cdots\!45$$$$\beta_{3} +$$$$19\!\cdots\!47$$$$\beta_{4} -$$$$83\!\cdots\!07$$$$\beta_{5} +$$$$19\!\cdots\!98$$$$\beta_{6} +$$$$78\!\cdots\!94$$$$\beta_{7} -$$$$28\!\cdots\!65$$$$\beta_{8} -$$$$26\!\cdots\!09$$$$\beta_{9} + 604479537347376095 \beta_{10} + 560341698491702539 \beta_{11} + 7398083809725978 \beta_{12} - 322530960064522 \beta_{13} - 30115526084926256 \beta_{14} - 29574679965329302 \beta_{15} - 8981198160985432 \beta_{16} + 7686685667906856 \beta_{17} - 75560632793478 \beta_{18} - 9253218719232 \beta_{19}) q^{38}$$ $$+($$$$27\!\cdots\!71$$$$-$$$$18\!\cdots\!85$$$$\beta_{1} -$$$$90\!\cdots\!69$$$$\beta_{2} -$$$$76\!\cdots\!45$$$$\beta_{3} -$$$$12\!\cdots\!32$$$$\beta_{4} +$$$$60\!\cdots\!32$$$$\beta_{5} -$$$$95\!\cdots\!23$$$$\beta_{6} -$$$$45\!\cdots\!73$$$$\beta_{7} -$$$$18\!\cdots\!07$$$$\beta_{8} + 15658614117589236391 \beta_{9} + 13684085684161320350 \beta_{10} + 1101929231445107559 \beta_{11} + 31197307485333938 \beta_{12} - 45882753554248143 \beta_{13} + 386085803024044149 \beta_{14} - 20720496663237683 \beta_{15} - 11117916377855755 \beta_{16} - 19114388289666397 \beta_{17} + 92459741647689 \beta_{18} - 14314753327095 \beta_{19}) q^{39}$$ $$+(-$$$$82\!\cdots\!70$$$$+$$$$34\!\cdots\!32$$$$\beta_{1} -$$$$37\!\cdots\!44$$$$\beta_{2} +$$$$10\!\cdots\!96$$$$\beta_{3} +$$$$37\!\cdots\!54$$$$\beta_{4} +$$$$70\!\cdots\!60$$$$\beta_{5} +$$$$85\!\cdots\!02$$$$\beta_{6} -$$$$84\!\cdots\!62$$$$\beta_{7} -$$$$15\!\cdots\!88$$$$\beta_{8} -$$$$10\!\cdots\!74$$$$\beta_{9} - 8499369835688002792 \beta_{10} + 5270851446735021512 \beta_{11} - 195488930021800376 \beta_{12} - 8320436061065798 \beta_{13} - 183773021170782682 \beta_{15} + 159832009611998646 \beta_{16}) q^{40}$$ $$+(-$$$$26\!\cdots\!04$$$$-$$$$41\!\cdots\!86$$$$\beta_{1} +$$$$13\!\cdots\!11$$$$\beta_{2} +$$$$72\!\cdots\!90$$$$\beta_{3} -$$$$17\!\cdots\!37$$$$\beta_{4} -$$$$45\!\cdots\!35$$$$\beta_{5} +$$$$13\!\cdots\!44$$$$\beta_{6} -$$$$99\!\cdots\!06$$$$\beta_{7} +$$$$82\!\cdots\!67$$$$\beta_{8} +$$$$13\!\cdots\!26$$$$\beta_{9} + 3423431548333403395 \beta_{10} + 3734462031573249496 \beta_{11} - 121378894588318257 \beta_{12} - 93847377761042256 \beta_{13} - 2608687009127923208 \beta_{14} + 151437049866236937 \beta_{15} + 5960235914252598 \beta_{16} - 257477902390076718 \beta_{17} + 841844375306886 \beta_{18} + 93374709914070 \beta_{19}) q^{41}$$ $$+(-$$$$18\!\cdots\!28$$$$+$$$$12\!\cdots\!39$$$$\beta_{1} +$$$$61\!\cdots\!59$$$$\beta_{2} +$$$$86\!\cdots\!07$$$$\beta_{3} +$$$$41\!\cdots\!39$$$$\beta_{4} -$$$$40\!\cdots\!71$$$$\beta_{5} -$$$$27\!\cdots\!94$$$$\beta_{6} -$$$$13\!\cdots\!34$$$$\beta_{7} -$$$$30\!\cdots\!43$$$$\beta_{8} +$$$$73\!\cdots\!76$$$$\beta_{9} - 13316069706343073545 \beta_{10} + 33921596468841078764 \beta_{11} - 1332397914148435298 \beta_{12} + 199329476970357588 \beta_{13} - 215577088867238432 \beta_{14} - 67089424568889298 \beta_{15} + 295166704849728052 \beta_{16} - 2187488270765392 \beta_{17} - 1502585694337172 \beta_{18} + 174549301330432 \beta_{19}) q^{42}$$ $$+($$$$14\!\cdots\!28$$$$-$$$$16\!\cdots\!61$$$$\beta_{1} +$$$$17\!\cdots\!44$$$$\beta_{2} +$$$$18\!\cdots\!27$$$$\beta_{3} -$$$$77\!\cdots\!87$$$$\beta_{4} +$$$$68\!\cdots\!07$$$$\beta_{5} -$$$$11\!\cdots\!20$$$$\beta_{6} +$$$$67\!\cdots\!03$$$$\beta_{7} -$$$$14\!\cdots\!71$$$$\beta_{8} -$$$$77\!\cdots\!08$$$$\beta_{9} +$$$$25\!\cdots\!15$$$$\beta_{10} + 56668649224536541412 \beta_{11} + 592196811077017277 \beta_{12} + 376646756131971362 \beta_{13} + 3020831181832291381 \beta_{15} - 989420785105561092 \beta_{16}) q^{43}$$ $$+($$$$20\!\cdots\!17$$$$-$$$$18\!\cdots\!98$$$$\beta_{1} -$$$$10\!\cdots\!97$$$$\beta_{2} -$$$$56\!\cdots\!46$$$$\beta_{3} +$$$$30\!\cdots\!74$$$$\beta_{4} -$$$$31\!\cdots\!05$$$$\beta_{5} +$$$$79\!\cdots\!39$$$$\beta_{6} +$$$$15\!\cdots\!79$$$$\beta_{7} +$$$$75\!\cdots\!17$$$$\beta_{8} +$$$$19\!\cdots\!87$$$$\beta_{9} +$$$$23\!\cdots\!47$$$$\beta_{10} +$$$$23\!\cdots\!84$$$$\beta_{11} + 734577223072904121 \beta_{12} + 1534408937685069513 \beta_{13} + 36857148901415351782 \beta_{14} + 2354484349189565974 \beta_{15} + 1345763797775248291 \beta_{16} + 2836712396227838114 \beta_{17} - 6631768701337088 \beta_{18} - 698446755210240 \beta_{19}) q^{44}$$ $$+($$$$14\!\cdots\!17$$$$+$$$$29\!\cdots\!25$$$$\beta_{1} -$$$$11\!\cdots\!92$$$$\beta_{2} -$$$$55\!\cdots\!45$$$$\beta_{3} +$$$$39\!\cdots\!62$$$$\beta_{4} -$$$$46\!\cdots\!31$$$$\beta_{5} -$$$$13\!\cdots\!72$$$$\beta_{6} +$$$$36\!\cdots\!12$$$$\beta_{7} -$$$$62\!\cdots\!15$$$$\beta_{8} -$$$$21\!\cdots\!74$$$$\beta_{9} -$$$$87\!\cdots\!15$$$$\beta_{10} +$$$$24\!\cdots\!86$$$$\beta_{11} + 18203294087304472385 \beta_{12} + 1505755454302869716 \beta_{13} - 40245516658730326416 \beta_{14} + 11376971343588290605 \beta_{15} - 1027867524435997192 \beta_{16} + 1696702787237689782 \beta_{17} + 16768080674269554 \beta_{18} - 1704123345500286 \beta_{19}) q^{45}$$ $$+(-$$$$46\!\cdots\!86$$$$+$$$$96\!\cdots\!40$$$$\beta_{1} -$$$$10\!\cdots\!40$$$$\beta_{2} +$$$$23\!\cdots\!66$$$$\beta_{3} +$$$$87\!\cdots\!94$$$$\beta_{4} +$$$$92\!\cdots\!82$$$$\beta_{5} -$$$$34\!\cdots\!16$$$$\beta_{6} +$$$$53\!\cdots\!20$$$$\beta_{7} -$$$$26\!\cdots\!90$$$$\beta_{8} +$$$$27\!\cdots\!02$$$$\beta_{9} +$$$$13\!\cdots\!86$$$$\beta_{10} +$$$$77\!\cdots\!14$$$$\beta_{11} + 11614594066073560800 \beta_{12} - 2182980601754876672 \beta_{13} - 11892437666856537888 \beta_{15} + 8106690016046478528 \beta_{16}) q^{46}$$ $$+($$$$18\!\cdots\!82$$$$+$$$$10\!\cdots\!50$$$$\beta_{1} -$$$$92\!\cdots\!66$$$$\beta_{2} -$$$$52\!\cdots\!92$$$$\beta_{3} -$$$$31\!\cdots\!34$$$$\beta_{4} -$$$$13\!\cdots\!00$$$$\beta_{5} +$$$$34\!\cdots\!16$$$$\beta_{6} -$$$$87\!\cdots\!14$$$$\beta_{7} -$$$$72\!\cdots\!92$$$$\beta_{8} -$$$$12\!\cdots\!08$$$$\beta_{9} +$$$$10\!\cdots\!60$$$$\beta_{10} +$$$$99\!\cdots\!98$$$$\beta_{11} - 3058790064395549874 \beta_{12} - 7641996230060385504 \beta_{13} -$$$$17\!\cdots\!22$$$$\beta_{14} - 14049308196535584494 \beta_{15} - 7395991619282240324 \beta_{16} - 13468563642446704348 \beta_{17} + 31338547099043884 \beta_{18} + 3296731963707756 \beta_{19}) q^{47}$$ $$+($$$$22\!\cdots\!59$$$$-$$$$99\!\cdots\!60$$$$\beta_{1} -$$$$18\!\cdots\!48$$$$\beta_{2} +$$$$61\!\cdots\!64$$$$\beta_{3} +$$$$12\!\cdots\!23$$$$\beta_{4} +$$$$21\!\cdots\!22$$$$\beta_{5} +$$$$28\!\cdots\!25$$$$\beta_{6} -$$$$28\!\cdots\!89$$$$\beta_{7} +$$$$18\!\cdots\!64$$$$\beta_{8} -$$$$83\!\cdots\!13$$$$\beta_{9} +$$$$15\!\cdots\!36$$$$\beta_{10} +$$$$32\!\cdots\!40$$$$\beta_{11} -$$$$15\!\cdots\!96$$$$\beta_{12} - 20557299137900844525 \beta_{13} +$$$$41\!\cdots\!92$$$$\beta_{14} -$$$$11\!\cdots\!67$$$$\beta_{15} + 2013731952482154429 \beta_{16} - 17175823382974874304 \beta_{17} - 135278111441676672 \beta_{18} + 13212709947873792 \beta_{19}) q^{48}$$ $$+($$$$17\!\cdots\!40$$$$+$$$$34\!\cdots\!14$$$$\beta_{1} -$$$$37\!\cdots\!74$$$$\beta_{2} -$$$$17\!\cdots\!45$$$$\beta_{3} +$$$$41\!\cdots\!20$$$$\beta_{4} +$$$$44\!\cdots\!99$$$$\beta_{5} +$$$$12\!\cdots\!42$$$$\beta_{6} -$$$$12\!\cdots\!49$$$$\beta_{7} -$$$$10\!\cdots\!50$$$$\beta_{8} +$$$$17\!\cdots\!03$$$$\beta_{9} -$$$$17\!\cdots\!39$$$$\beta_{10} +$$$$34\!\cdots\!85$$$$\beta_{11} -$$$$20\!\cdots\!15$$$$\beta_{12} + 6361926049382228794 \beta_{13} - 26382667440693627714 \beta_{15} - 70650975683798855541 \beta_{16}) q^{49}$$ $$+($$$$10\!\cdots\!50$$$$+$$$$10\!\cdots\!25$$$$\beta_{1} -$$$$54\!\cdots\!00$$$$\beta_{2} -$$$$31\!\cdots\!00$$$$\beta_{3} -$$$$40\!\cdots\!50$$$$\beta_{4} -$$$$52\!\cdots\!50$$$$\beta_{5} +$$$$12\!\cdots\!50$$$$\beta_{6} -$$$$30\!\cdots\!50$$$$\beta_{7} -$$$$52\!\cdots\!00$$$$\beta_{8} -$$$$10\!\cdots\!50$$$$\beta_{9} +$$$$37\!\cdots\!00$$$$\beta_{10} +$$$$36\!\cdots\!00$$$$\beta_{11} + 4083867489089985150 \beta_{12} - 32295314607163363950 \beta_{13} -$$$$71\!\cdots\!00$$$$\beta_{14} -$$$$12\!\cdots\!50$$$$\beta_{15} - 50882821331250104600 \beta_{16} - 38241092589882317800 \beta_{17} + 14961606940315150 \beta_{18} + 108488553206400 \beta_{19}) q^{50}$$ $$+($$$$12\!\cdots\!83$$$$-$$$$19\!\cdots\!83$$$$\beta_{1} -$$$$16\!\cdots\!22$$$$\beta_{2} -$$$$42\!\cdots\!56$$$$\beta_{3} +$$$$87\!\cdots\!92$$$$\beta_{4} -$$$$50\!\cdots\!63$$$$\beta_{5} +$$$$50\!\cdots\!60$$$$\beta_{6} +$$$$15\!\cdots\!21$$$$\beta_{7} +$$$$52\!\cdots\!41$$$$\beta_{8} -$$$$28\!\cdots\!04$$$$\beta_{9} -$$$$20\!\cdots\!15$$$$\beta_{10} -$$$$26\!\cdots\!55$$$$\beta_{11} +$$$$79\!\cdots\!46$$$$\beta_{12} + 20981125375805048586 \beta_{13} -$$$$11\!\cdots\!91$$$$\beta_{14} +$$$$30\!\cdots\!20$$$$\beta_{15} -$$$$10\!\cdots\!54$$$$\beta_{16} + 57686974309737340162 \beta_{17} + 755044498853112326 \beta_{18} - 77812741139237050 \beta_{19}) q^{51}$$ $$+(-$$$$35\!\cdots\!76$$$$-$$$$18\!\cdots\!10$$$$\beta_{1} +$$$$19\!\cdots\!74$$$$\beta_{2} +$$$$18\!\cdots\!34$$$$\beta_{3} -$$$$22\!\cdots\!68$$$$\beta_{4} -$$$$54\!\cdots\!40$$$$\beta_{5} -$$$$20\!\cdots\!20$$$$\beta_{6} +$$$$86\!\cdots\!60$$$$\beta_{7} -$$$$48\!\cdots\!88$$$$\beta_{8} +$$$$12\!\cdots\!56$$$$\beta_{9} +$$$$11\!\cdots\!40$$$$\beta_{10} -$$$$20\!\cdots\!64$$$$\beta_{11} +$$$$16\!\cdots\!96$$$$\beta_{12} -$$$$13\!\cdots\!64$$$$\beta_{13} -$$$$29\!\cdots\!12$$$$\beta_{15} +$$$$41\!\cdots\!84$$$$\beta_{16}) q^{52}$$ $$+($$$$31\!\cdots\!09$$$$-$$$$56\!\cdots\!71$$$$\beta_{1} -$$$$15\!\cdots\!67$$$$\beta_{2} -$$$$89\!\cdots\!89$$$$\beta_{3} -$$$$32\!\cdots\!13$$$$\beta_{4} -$$$$17\!\cdots\!30$$$$\beta_{5} +$$$$26\!\cdots\!12$$$$\beta_{6} +$$$$58\!\cdots\!92$$$$\beta_{7} +$$$$34\!\cdots\!06$$$$\beta_{8} +$$$$86\!\cdots\!84$$$$\beta_{9} +$$$$22\!\cdots\!30$$$$\beta_{10} +$$$$32\!\cdots\!36$$$$\beta_{11} +$$$$25\!\cdots\!42$$$$\beta_{12} +$$$$58\!\cdots\!52$$$$\beta_{13} +$$$$14\!\cdots\!96$$$$\beta_{14} +$$$$99\!\cdots\!82$$$$\beta_{15} +$$$$54\!\cdots\!52$$$$\beta_{16} +$$$$10\!\cdots\!44$$$$\beta_{17} - 1850377327283043572 \beta_{18} - 179313044337464148 \beta_{19}) q^{53}$$ $$+(-$$$$19\!\cdots\!97$$$$+$$$$48\!\cdots\!49$$$$\beta_{1} +$$$$69\!\cdots\!05$$$$\beta_{2} -$$$$81\!\cdots\!53$$$$\beta_{3} +$$$$68\!\cdots\!02$$$$\beta_{4} -$$$$67\!\cdots\!26$$$$\beta_{5} -$$$$11\!\cdots\!42$$$$\beta_{6} +$$$$45\!\cdots\!74$$$$\beta_{7} -$$$$90\!\cdots\!41$$$$\beta_{8} +$$$$19\!\cdots\!97$$$$\beta_{9} -$$$$49\!\cdots\!45$$$$\beta_{10} -$$$$74\!\cdots\!91$$$$\beta_{11} -$$$$10\!\cdots\!60$$$$\beta_{12} +$$$$10\!\cdots\!74$$$$\beta_{13} -$$$$10\!\cdots\!72$$$$\beta_{14} +$$$$18\!\cdots\!88$$$$\beta_{15} +$$$$74\!\cdots\!00$$$$\beta_{16} +$$$$37\!\cdots\!44$$$$\beta_{17} - 2112081418711566918 \beta_{18} + 292454857626024960 \beta_{19}) q^{54}$$ $$+($$$$65\!\cdots\!70$$$$+$$$$86\!\cdots\!40$$$$\beta_{1} -$$$$92\!\cdots\!33$$$$\beta_{2} -$$$$49\!\cdots\!64$$$$\beta_{3} +$$$$12\!\cdots\!51$$$$\beta_{4} -$$$$22\!\cdots\!85$$$$\beta_{5} +$$$$17\!\cdots\!16$$$$\beta_{6} -$$$$27\!\cdots\!60$$$$\beta_{7} +$$$$79\!\cdots\!31$$$$\beta_{8} -$$$$34\!\cdots\!72$$$$\beta_{9} -$$$$76\!\cdots\!35$$$$\beta_{10} -$$$$20\!\cdots\!64$$$$\beta_{11} -$$$$79\!\cdots\!53$$$$\beta_{12} +$$$$19\!\cdots\!06$$$$\beta_{13} +$$$$92\!\cdots\!79$$$$\beta_{15} -$$$$29\!\cdots\!12$$$$\beta_{16}) q^{55}$$ $$+(-$$$$43\!\cdots\!14$$$$-$$$$67\!\cdots\!56$$$$\beta_{1} +$$$$21\!\cdots\!04$$$$\beta_{2} +$$$$11\!\cdots\!74$$$$\beta_{3} -$$$$35\!\cdots\!46$$$$\beta_{4} +$$$$74\!\cdots\!22$$$$\beta_{5} -$$$$17\!\cdots\!00$$$$\beta_{6} -$$$$33\!\cdots\!74$$$$\beta_{7} +$$$$17\!\cdots\!94$$$$\beta_{8} +$$$$30\!\cdots\!20$$$$\beta_{9} -$$$$54\!\cdots\!28$$$$\beta_{10} -$$$$53\!\cdots\!12$$$$\beta_{11} -$$$$32\!\cdots\!64$$$$\beta_{12} -$$$$27\!\cdots\!08$$$$\beta_{13} -$$$$77\!\cdots\!68$$$$\beta_{14} +$$$$33\!\cdots\!44$$$$\beta_{15} -$$$$15\!\cdots\!28$$$$\beta_{16} -$$$$72\!\cdots\!48$$$$\beta_{17} + 19167543270266925696 \beta_{18} + 2005471050215109120 \beta_{19}) q^{56}$$ $$+($$$$72\!\cdots\!09$$$$+$$$$91\!\cdots\!25$$$$\beta_{1} +$$$$12\!\cdots\!74$$$$\beta_{2} -$$$$26\!\cdots\!56$$$$\beta_{3} +$$$$34\!\cdots\!62$$$$\beta_{4} +$$$$85\!\cdots\!88$$$$\beta_{5} -$$$$13\!\cdots\!32$$$$\beta_{6} -$$$$12\!\cdots\!74$$$$\beta_{7} -$$$$15\!\cdots\!86$$$$\beta_{8} +$$$$16\!\cdots\!19$$$$\beta_{9} +$$$$21\!\cdots\!07$$$$\beta_{10} -$$$$50\!\cdots\!78$$$$\beta_{11} -$$$$22\!\cdots\!49$$$$\beta_{12} -$$$$87\!\cdots\!52$$$$\beta_{13} +$$$$10\!\cdots\!40$$$$\beta_{14} -$$$$12\!\cdots\!49$$$$\beta_{15} +$$$$22\!\cdots\!64$$$$\beta_{16} -$$$$55\!\cdots\!94$$$$\beta_{17} - 9079817399649036126 \beta_{18} + 86086161768973266 \beta_{19}) q^{57}$$ $$+(-$$$$50\!\cdots\!68$$$$-$$$$88\!\cdots\!43$$$$\beta_{1} +$$$$94\!\cdots\!73$$$$\beta_{2} +$$$$77\!\cdots\!19$$$$\beta_{3} -$$$$22\!\cdots\!31$$$$\beta_{4} -$$$$80\!\cdots\!65$$$$\beta_{5} +$$$$47\!\cdots\!10$$$$\beta_{6} +$$$$14\!\cdots\!98$$$$\beta_{7} +$$$$14\!\cdots\!83$$$$\beta_{8} +$$$$15\!\cdots\!04$$$$\beta_{9} -$$$$38\!\cdots\!43$$$$\beta_{10} -$$$$64\!\cdots\!60$$$$\beta_{11} +$$$$98\!\cdots\!44$$$$\beta_{12} -$$$$12\!\cdots\!60$$$$\beta_{13} -$$$$66\!\cdots\!08$$$$\beta_{15} +$$$$34\!\cdots\!44$$$$\beta_{16}) q^{58}$$ $$+($$$$35\!\cdots\!78$$$$+$$$$19\!\cdots\!77$$$$\beta_{1} -$$$$17\!\cdots\!75$$$$\beta_{2} -$$$$99\!\cdots\!89$$$$\beta_{3} -$$$$74\!\cdots\!32$$$$\beta_{4} +$$$$87\!\cdots\!54$$$$\beta_{5} -$$$$24\!\cdots\!36$$$$\beta_{6} +$$$$13\!\cdots\!39$$$$\beta_{7} -$$$$83\!\cdots\!20$$$$\beta_{8} -$$$$42\!\cdots\!80$$$$\beta_{9} -$$$$66\!\cdots\!96$$$$\beta_{10} -$$$$74\!\cdots\!60$$$$\beta_{11} +$$$$14\!\cdots\!64$$$$\beta_{12} +$$$$22\!\cdots\!00$$$$\beta_{13} +$$$$27\!\cdots\!76$$$$\beta_{14} -$$$$47\!\cdots\!44$$$$\beta_{15} -$$$$13\!\cdots\!04$$$$\beta_{16} +$$$$18\!\cdots\!52$$$$\beta_{17} -$$$$11\!\cdots\!44$$$$\beta_{18} - 13482718801134233400 \beta_{19}) q^{59}$$ $$+($$$$94\!\cdots\!47$$$$-$$$$13\!\cdots\!78$$$$\beta_{1} -$$$$74\!\cdots\!31$$$$\beta_{2} -$$$$24\!\cdots\!22$$$$\beta_{3} +$$$$11\!\cdots\!90$$$$\beta_{4} +$$$$25\!\cdots\!81$$$$\beta_{5} +$$$$13\!\cdots\!69$$$$\beta_{6} +$$$$63\!\cdots\!09$$$$\beta_{7} +$$$$31\!\cdots\!67$$$$\beta_{8} -$$$$19\!\cdots\!95$$$$\beta_{9} -$$$$90\!\cdots\!59$$$$\beta_{10} -$$$$18\!\cdots\!64$$$$\beta_{11} +$$$$21\!\cdots\!59$$$$\beta_{12} +$$$$27\!\cdots\!31$$$$\beta_{13} -$$$$29\!\cdots\!14$$$$\beta_{14} +$$$$17\!\cdots\!98$$$$\beta_{15} -$$$$47\!\cdots\!87$$$$\beta_{16} +$$$$26\!\cdots\!38$$$$\beta_{17} +$$$$16\!\cdots\!96$$$$\beta_{18} - 12220617046878639104 \beta_{19}) q^{60}$$ $$+($$$$18\!\cdots\!55$$$$-$$$$73\!\cdots\!66$$$$\beta_{1} +$$$$78\!\cdots\!74$$$$\beta_{2} +$$$$14\!\cdots\!65$$$$\beta_{3} -$$$$10\!\cdots\!41$$$$\beta_{4} -$$$$25\!\cdots\!03$$$$\beta_{5} -$$$$44\!\cdots\!36$$$$\beta_{6} +$$$$19\!\cdots\!85$$$$\beta_{7} +$$$$43\!\cdots\!90$$$$\beta_{8} +$$$$36\!\cdots\!92$$$$\beta_{9} -$$$$31\!\cdots\!39$$$$\beta_{10} -$$$$18\!\cdots\!11$$$$\beta_{11} +$$$$11\!\cdots\!40$$$$\beta_{12} +$$$$26\!\cdots\!03$$$$\beta_{13} +$$$$22\!\cdots\!07$$$$\beta_{15} -$$$$28\!\cdots\!82$$$$\beta_{16}) q^{61}$$ $$+($$$$44\!\cdots\!40$$$$+$$$$18\!\cdots\!14$$$$\beta_{1} -$$$$22\!\cdots\!72$$$$\beta_{2} -$$$$12\!\cdots\!55$$$$\beta_{3} +$$$$22\!\cdots\!44$$$$\beta_{4} -$$$$81\!\cdots\!32$$$$\beta_{5} +$$$$19\!\cdots\!49$$$$\beta_{6} +$$$$22\!\cdots\!09$$$$\beta_{7} -$$$$38\!\cdots\!11$$$$\beta_{8} +$$$$34\!\cdots\!38$$$$\beta_{9} +$$$$60\!\cdots\!37$$$$\beta_{10} +$$$$60\!\cdots\!91$$$$\beta_{11} +$$$$26\!\cdots\!53$$$$\beta_{12} +$$$$40\!\cdots\!67$$$$\beta_{13} +$$$$18\!\cdots\!48$$$$\beta_{14} +$$$$33\!\cdots\!81$$$$\beta_{15} +$$$$27\!\cdots\!68$$$$\beta_{16} +$$$$83\!\cdots\!20$$$$\beta_{17} +$$$$40\!\cdots\!01$$$$\beta_{18} + 57775891714146952704 \beta_{19}) q^{62}$$ $$+($$$$67\!\cdots\!16$$$$-$$$$87\!\cdots\!23$$$$\beta_{1} +$$$$55\!\cdots\!39$$$$\beta_{2} -$$$$86\!\cdots\!64$$$$\beta_{3} +$$$$18\!\cdots\!31$$$$\beta_{4} -$$$$94\!\cdots\!72$$$$\beta_{5} +$$$$47\!\cdots\!13$$$$\beta_{6} +$$$$81\!\cdots\!04$$$$\beta_{7} +$$$$10\!\cdots\!94$$$$\beta_{8} +$$$$35\!\cdots\!62$$$$\beta_{9} -$$$$47\!\cdots\!92$$$$\beta_{10} +$$$$16\!\cdots\!80$$$$\beta_{11} -$$$$97\!\cdots\!70$$$$\beta_{12} +$$$$71\!\cdots\!82$$$$\beta_{13} -$$$$12\!\cdots\!96$$$$\beta_{14} -$$$$19\!\cdots\!32$$$$\beta_{15} +$$$$15\!\cdots\!54$$$$\beta_{16} -$$$$34\!\cdots\!46$$$$\beta_{17} -$$$$11\!\cdots\!66$$$$\beta_{18} +$$$$11\!\cdots\!66$$$$\beta_{19}) q^{63}$$ $$+(-$$$$21\!\cdots\!36$$$$+$$$$15\!\cdots\!76$$$$\beta_{1} -$$$$16\!\cdots\!44$$$$\beta_{2} +$$$$41\!\cdots\!16$$$$\beta_{3} +$$$$11\!\cdots\!12$$$$\beta_{4} +$$$$49\!\cdots\!96$$$$\beta_{5} -$$$$67\!\cdots\!00$$$$\beta_{6} +$$$$36\!\cdots\!76$$$$\beta_{7} -$$$$10\!\cdots\!64$$$$\beta_{8} -$$$$42\!\cdots\!16$$$$\beta_{9} +$$$$18\!\cdots\!72$$$$\beta_{10} +$$$$38\!\cdots\!28$$$$\beta_{11} -$$$$53\!\cdots\!16$$$$\beta_{12} +$$$$10\!\cdots\!00$$$$\beta_{13} -$$$$50\!\cdots\!60$$$$\beta_{15} +$$$$13\!\cdots\!68$$$$\beta_{16}) q^{64}$$ $$+(-$$$$13\!\cdots\!74$$$$-$$$$17\!\cdots\!86$$$$\beta_{1} +$$$$65\!\cdots\!71$$$$\beta_{2} +$$$$36\!\cdots\!28$$$$\beta_{3} +$$$$18\!\cdots\!31$$$$\beta_{4} -$$$$40\!\cdots\!97$$$$\beta_{5} +$$$$10\!\cdots\!60$$$$\beta_{6} -$$$$58\!\cdots\!96$$$$\beta_{7} +$$$$66\!\cdots\!79$$$$\beta_{8} +$$$$93\!\cdots\!64$$$$\beta_{9} +$$$$29\!\cdots\!45$$$$\beta_{10} +$$$$31\!\cdots\!66$$$$\beta_{11} -$$$$53\!\cdots\!67$$$$\beta_{12} -$$$$29\!\cdots\!64$$$$\beta_{13} -$$$$99\!\cdots\!72$$$$\beta_{14} +$$$$10\!\cdots\!61$$$$\beta_{15} +$$$$19\!\cdots\!28$$$$\beta_{16} -$$$$97\!\cdots\!26$$$$\beta_{17} + 42293463504303299158 \beta_{18} - 86790530492642506842 \beta_{19}) q^{65}$$ $$+($$$$41\!\cdots\!39$$$$+$$$$29\!\cdots\!55$$$$\beta_{1} +$$$$59\!\cdots\!74$$$$\beta_{2} -$$$$11\!\cdots\!39$$$$\beta_{3} -$$$$55\!\cdots\!27$$$$\beta_{4} -$$$$21\!\cdots\!27$$$$\beta_{5} +$$$$22\!\cdots\!69$$$$\beta_{6} -$$$$13\!\cdots\!91$$$$\beta_{7} -$$$$64\!\cdots\!82$$$$\beta_{8} +$$$$41\!\cdots\!09$$$$\beta_{9} +$$$$19\!\cdots\!34$$$$\beta_{10} +$$$$55\!\cdots\!14$$$$\beta_{11} +$$$$24\!\cdots\!30$$$$\beta_{12} -$$$$14\!\cdots\!25$$$$\beta_{13} +$$$$11\!\cdots\!84$$$$\beta_{14} +$$$$26\!\cdots\!18$$$$\beta_{15} +$$$$40\!\cdots\!14$$$$\beta_{16} -$$$$71\!\cdots\!84$$$$\beta_{17} +$$$$57\!\cdots\!53$$$$\beta_{18} -$$$$75\!\cdots\!80$$$$\beta_{19}) q^{66}$$ $$+(-$$$$31\!\cdots\!50$$$$+$$$$19\!\cdots\!35$$$$\beta_{1} -$$$$20\!\cdots\!07$$$$\beta_{2} +$$$$10\!\cdots\!99$$$$\beta_{3} +$$$$60\!\cdots\!66$$$$\beta_{4} +$$$$48\!\cdots\!64$$$$\beta_{5} +$$$$16\!\cdots\!96$$$$\beta_{6} +$$$$21\!\cdots\!31$$$$\beta_{7} -$$$$13\!\cdots\!94$$$$\beta_{8} +$$$$14\!\cdots\!88$$$$\beta_{9} -$$$$61\!\cdots\!66$$$$\beta_{10} +$$$$50\!\cdots\!00$$$$\beta_{11} -$$$$13\!\cdots\!02$$$$\beta_{12} -$$$$10\!\cdots\!00$$$$\beta_{13} +$$$$26\!\cdots\!34$$$$\beta_{15} -$$$$26\!\cdots\!32$$$$\beta_{16}) q^{67}$$ $$+($$$$27\!\cdots\!08$$$$-$$$$63\!\cdots\!04$$$$\beta_{1} -$$$$13\!\cdots\!04$$$$\beta_{2} -$$$$75\!\cdots\!00$$$$\beta_{3} +$$$$17\!\cdots\!72$$$$\beta_{4} -$$$$17\!\cdots\!68$$$$\beta_{5} +$$$$40\!\cdots\!36$$$$\beta_{6} +$$$$45\!\cdots\!44$$$$\beta_{7} +$$$$13\!\cdots\!88$$$$\beta_{8} -$$$$29\!\cdots\!68$$$$\beta_{9} +$$$$12\!\cdots\!28$$$$\beta_{10} +$$$$11\!\cdots\!64$$$$\beta_{11} +$$$$19\!\cdots\!72$$$$\beta_{12} +$$$$12\!\cdots\!08$$$$\beta_{13} +$$$$93\!\cdots\!92$$$$\beta_{14} -$$$$30\!\cdots\!36$$$$\beta_{15} -$$$$39\!\cdots\!08$$$$\beta_{16} +$$$$37\!\cdots\!20$$$$\beta_{17} -$$$$11\!\cdots\!76$$$$\beta_{18} -$$$$96\!\cdots\!64$$$$\beta_{19}) q^{68}$$ $$+($$$$13\!\cdots\!44$$$$-$$$$26\!\cdots\!74$$$$\beta_{1} +$$$$88\!\cdots\!19$$$$\beta_{2} -$$$$21\!\cdots\!56$$$$\beta_{3} -$$$$10\!\cdots\!85$$$$\beta_{4} -$$$$42\!\cdots\!45$$$$\beta_{5} -$$$$25\!\cdots\!44$$$$\beta_{6} +$$$$78\!\cdots\!70$$$$\beta_{7} -$$$$31\!\cdots\!55$$$$\beta_{8} -$$$$27\!\cdots\!24$$$$\beta_{9} +$$$$11\!\cdots\!67$$$$\beta_{10} +$$$$38\!\cdots\!32$$$$\beta_{11} -$$$$39\!\cdots\!69$$$$\beta_{12} +$$$$10\!\cdots\!96$$$$\beta_{13} -$$$$14\!\cdots\!96$$$$\beta_{14} -$$$$12\!\cdots\!87$$$$\beta_{15} -$$$$31\!\cdots\!70$$$$\beta_{16} +$$$$44\!\cdots\!18$$$$\beta_{17} -$$$$19\!\cdots\!06$$$$\beta_{18} +$$$$35\!\cdots\!70$$$$\beta_{19}) q^{69}$$ $$+(-$$$$26\!\cdots\!90$$$$-$$$$18\!\cdots\!60$$$$\beta_{1} +$$$$19\!\cdots\!06$$$$\beta_{2} +$$$$73\!\cdots\!78$$$$\beta_{3} -$$$$31\!\cdots\!72$$$$\beta_{4} -$$$$34\!\cdots\!80$$$$\beta_{5} +$$$$26\!\cdots\!28$$$$\beta_{6} +$$$$93\!\cdots\!00$$$$\beta_{7} +$$$$44\!\cdots\!48$$$$\beta_{8} +$$$$80\!\cdots\!74$$$$\beta_{9} -$$$$34\!\cdots\!00$$$$\beta_{10} -$$$$26\!\cdots\!62$$$$\beta_{11} +$$$$22\!\cdots\!26$$$$\beta_{12} +$$$$35\!\cdots\!48$$$$\beta_{13} -$$$$11\!\cdots\!18$$$$\beta_{15} -$$$$54\!\cdots\!96$$$$\beta_{16}) q^{70}$$ $$+(-$$$$11\!\cdots\!88$$$$+$$$$55\!\cdots\!84$$$$\beta_{1} +$$$$56\!\cdots\!43$$$$\beta_{2} +$$$$31\!\cdots\!42$$$$\beta_{3} +$$$$35\!\cdots\!17$$$$\beta_{4} +$$$$10\!\cdots\!23$$$$\beta_{5} -$$$$26\!\cdots\!04$$$$\beta_{6} +$$$$66\!\cdots\!50$$$$\beta_{7} -$$$$16\!\cdots\!11$$$$\beta_{8} -$$$$84\!\cdots\!02$$$$\beta_{9} -$$$$75\!\cdots\!59$$$$\beta_{10} -$$$$77\!\cdots\!22$$$$\beta_{11} +$$$$87\!\cdots\!63$$$$\beta_{12} -$$$$22\!\cdots\!80$$$$\beta_{13} +$$$$11\!\cdots\!06$$$$\beta_{14} -$$$$11\!\cdots\!23$$$$\beta_{15} -$$$$42\!\cdots\!98$$$$\beta_{16} -$$$$19\!\cdots\!66$$$$\beta_{17} +$$$$87\!\cdots\!02$$$$\beta_{18} +$$$$10\!\cdots\!90$$$$\beta_{19}) q^{71}$$ $$+($$$$67\!\cdots\!83$$$$-$$$$35\!\cdots\!32$$$$\beta_{1} +$$$$46\!\cdots\!42$$$$\beta_{2} -$$$$21\!\cdots\!41$$$$\beta_{3} -$$$$33\!\cdots\!91$$$$\beta_{4} +$$$$10\!\cdots\!87$$$$\beta_{5} -$$$$17\!\cdots\!10$$$$\beta_{6} +$$$$38\!\cdots\!25$$$$\beta_{7} +$$$$89\!\cdots\!97$$$$\beta_{8} +$$$$44\!\cdots\!74$$$$\beta_{9} -$$$$80\!\cdots\!88$$$$\beta_{10} -$$$$16\!\cdots\!40$$$$\beta_{11} +$$$$24\!\cdots\!48$$$$\beta_{12} -$$$$40\!\cdots\!10$$$$\beta_{13} -$$$$17\!\cdots\!76$$$$\beta_{14} +$$$$14\!\cdots\!86$$$$\beta_{15} -$$$$38\!\cdots\!42$$$$\beta_{16} -$$$$10\!\cdots\!28$$$$\beta_{17} +$$$$40\!\cdots\!96$$$$\beta_{18} -$$$$12\!\cdots\!76$$$$\beta_{19}) q^{72}$$ $$+(-$$$$11\!\cdots\!82$$$$+$$$$42\!\cdots\!12$$$$\beta_{1} -$$$$45\!\cdots\!36$$$$\beta_{2} +$$$$13\!\cdots\!72$$$$\beta_{3} +$$$$22\!\cdots\!02$$$$\beta_{4} -$$$$14\!\cdots\!40$$$$\beta_{5} +$$$$12\!\cdots\!88$$$$\beta_{6} -$$$$23\!\cdots\!16$$$$\beta_{7} +$$$$52\!\cdots\!88$$$$\beta_{8} -$$$$24\!\cdots\!86$$$$\beta_{9} +$$$$89\!\cdots\!68$$$$\beta_{10} -$$$$14\!\cdots\!52$$$$\beta_{11} -$$$$89\!\cdots\!66$$$$\beta_{12} +$$$$10\!\cdots\!98$$$$\beta_{13} -$$$$79\!\cdots\!58$$$$\beta_{15} +$$$$41\!\cdots\!18$$$$\beta_{16}) q^{73}$$ $$+(-$$$$18\!\cdots\!66$$$$+$$$$18\!\cdots\!50$$$$\beta_{1} +$$$$94\!\cdots\!40$$$$\beta_{2} +$$$$52\!\cdots\!44$$$$\beta_{3} -$$$$98\!\cdots\!90$$$$\beta_{4} +$$$$24\!\cdots\!14$$$$\beta_{5} -$$$$64\!\cdots\!46$$$$\beta_{6} -$$$$19\!\cdots\!62$$$$\beta_{7} -$$$$34\!\cdots\!20$$$$\beta_{8} +$$$$48\!\cdots\!58$$$$\beta_{9} -$$$$20\!\cdots\!44$$$$\beta_{10} -$$$$19\!\cdots\!88$$$$\beta_{11} -$$$$24\!\cdots\!26$$$$\beta_{12} -$$$$11\!\cdots\!70$$$$\beta_{13} -$$$$40\!\cdots\!24$$$$\beta_{14} +$$$$54\!\cdots\!26$$$$\beta_{15} +$$$$11\!\cdots\!16$$$$\beta_{16} -$$$$42\!\cdots\!68$$$$\beta_{17} -$$$$40\!\cdots\!34$$$$\beta_{18} -$$$$59\!\cdots\!20$$$$\beta_{19}) q^{74}$$ $$+($$$$61\!\cdots\!55$$$$+$$$$16\!\cdots\!70$$$$\beta_{1} +$$$$15\!\cdots\!78$$$$\beta_{2} -$$$$93\!\cdots\!86$$$$\beta_{3} +$$$$15\!\cdots\!39$$$$\beta_{4} +$$$$34\!\cdots\!10$$$$\beta_{5} +$$$$23\!\cdots\!14$$$$\beta_{6} +$$$$67\!\cdots\!25$$$$\beta_{7} +$$$$75\!\cdots\!24$$$$\beta_{8} +$$$$15\!\cdots\!62$$$$\beta_{9} +$$$$29\!\cdots\!00$$$$\beta_{10} -$$$$10\!\cdots\!31$$$$\beta_{11} -$$$$21\!\cdots\!87$$$$\beta_{12} +$$$$67\!\cdots\!74$$$$\beta_{13} +$$$$94\!\cdots\!25$$$$\beta_{14} +$$$$90\!\cdots\!41$$$$\beta_{15} +$$$$78\!\cdots\!52$$$$\beta_{16} -$$$$21\!\cdots\!00$$$$\beta_{17} -$$$$43\!\cdots\!00$$$$\beta_{18} +$$$$29\!\cdots\!00$$$$\beta_{19}) q^{75}$$ $$+(-$$$$44\!\cdots\!21$$$$-$$$$78\!\cdots\!34$$$$\beta_{1} +$$$$84\!\cdots\!33$$$$\beta_{2} +$$$$14\!\cdots\!76$$$$\beta_{3} -$$$$88\!\cdots\!36$$$$\beta_{4} -$$$$64\!\cdots\!19$$$$\beta_{5} -$$$$12\!\cdots\!67$$$$\beta_{6} +$$$$29\!\cdots\!15$$$$\beta_{7} +$$$$12\!\cdots\!19$$$$\beta_{8} -$$$$11\!\cdots\!53$$$$\beta_{9} -$$$$28\!\cdots\!27$$$$\beta_{10} -$$$$32\!\cdots\!32$$$$\beta_{11} +$$$$10\!\cdots\!01$$$$\beta_{12} -$$$$10\!\cdots\!19$$$$\beta_{13} +$$$$17\!\cdots\!24$$$$\beta_{15} -$$$$59\!\cdots\!47$$$$\beta_{16}) q^{76}$$ $$+($$$$19\!\cdots\!54$$$$-$$$$39\!\cdots\!62$$$$\beta_{1} -$$$$99\!\cdots\!32$$$$\beta_{2} -$$$$55\!\cdots\!14$$$$\beta_{3} -$$$$65\!\cdots\!68$$$$\beta_{4} +$$$$24\!\cdots\!30$$$$\beta_{5} -$$$$96\!\cdots\!08$$$$\beta_{6} +$$$$42\!\cdots\!32$$$$\beta_{7} +$$$$10\!\cdots\!26$$$$\beta_{8} +$$$$77\!\cdots\!44$$$$\beta_{9} -$$$$29\!\cdots\!50$$$$\beta_{10} -$$$$28\!\cdots\!24$$$$\beta_{11} +$$$$36\!\cdots\!42$$$$\beta_{12} +$$$$77\!\cdots\!52$$$$\beta_{13} -$$$$93\!\cdots\!84$$$$\beta_{14} +$$$$11\!\cdots\!42$$$$\beta_{15} +$$$$67\!\cdots\!32$$$$\beta_{16} +$$$$14\!\cdots\!64$$$$\beta_{17} +$$$$12\!\cdots\!28$$$$\beta_{18} +$$$$22\!\cdots\!72$$$$\beta_{19}) q^{77}$$ $$+($$$$21\!\cdots\!34$$$$+$$$$54\!\cdots\!22$$$$\beta_{1} +$$$$33\!\cdots\!04$$$$\beta_{2} +$$$$16\!\cdots\!44$$$$\beta_{3} +$$$$10\!\cdots\!10$$$$\beta_{4} -$$$$73\!\cdots\!38$$$$\beta_{5} +$$$$64\!\cdots\!26$$$$\beta_{6} +$$$$71\!\cdots\!46$$$$\beta_{7} -$$$$11\!\cdots\!94$$$$\beta_{8} -$$$$73\!\cdots\!38$$$$\beta_{9} +$$$$65\!\cdots\!22$$$$\beta_{10} +$$$$51\!\cdots\!36$$$$\beta_{11} +$$$$11\!\cdots\!24$$$$\beta_{12} +$$$$28\!\cdots\!10$$$$\beta_{13} -$$$$88\!\cdots\!96$$$$\beta_{14} +$$$$37\!\cdots\!36$$$$\beta_{15} -$$$$22\!\cdots\!40$$$$\beta_{16} +$$$$24\!\cdots\!96$$$$\beta_{17} +$$$$18\!\cdots\!82$$$$\beta_{18} -$$$$22\!\cdots\!92$$$$\beta_{19}) q^{78}$$ $$+(-$$$$90\!\cdots\!66$$$$-$$$$16\!\cdots\!81$$$$\beta_{1} +$$$$17\!\cdots\!62$$$$\beta_{2} -$$$$10\!\cdots\!70$$$$\beta_{3} -$$$$16\!\cdots\!28$$$$\beta_{4} +$$$$20\!\cdots\!37$$$$\beta_{5} +$$$$86\!\cdots\!21$$$$\beta_{6} +$$$$86\!\cdots\!36$$$$\beta_{7} -$$$$64\!\cdots\!03$$$$\beta_{8} +$$$$23\!\cdots\!44$$$$\beta_{9} -$$$$52\!\cdots\!85$$$$\beta_{10} +$$$$20\!\cdots\!84$$$$\beta_{11} +$$$$31\!\cdots\!53$$$$\beta_{12} +$$$$71\!\cdots\!42$$$$\beta_{13} -$$$$28\!\cdots\!27$$$$\beta_{15} -$$$$86\!\cdots\!52$$$$\beta_{16}) q^{79}$$ $$+($$$$12\!\cdots\!76$$$$-$$$$13\!\cdots\!16$$$$\beta_{1} -$$$$60\!\cdots\!44$$$$\beta_{2} -$$$$33\!\cdots\!12$$$$\beta_{3} -$$$$46\!\cdots\!84$$$$\beta_{4} -$$$$80\!\cdots\!52$$$$\beta_{5} +$$$$18\!\cdots\!00$$$$\beta_{6} +$$$$18\!\cdots\!64$$$$\beta_{7} +$$$$24\!\cdots\!24$$$$\beta_{8} -$$$$72\!\cdots\!96$$$$\beta_{9} +$$$$56\!\cdots\!60$$$$\beta_{10} +$$$$54\!\cdots\!96$$$$\beta_{11} +$$$$26\!\cdots\!68$$$$\beta_{12} +$$$$35\!\cdots\!36$$$$\beta_{13} +$$$$11\!\cdots\!68$$$$\beta_{14} -$$$$89\!\cdots\!04$$$$\beta_{15} -$$$$25\!\cdots\!92$$$$\beta_{16} +$$$$33\!\cdots\!64$$$$\beta_{17} -$$$$23\!\cdots\!92$$$$\beta_{18} -$$$$55\!\cdots\!72$$$$\beta_{19}) q^{80}$$ $$+(-$$$$13\!\cdots\!20$$$$+$$$$36\!\cdots\!12$$$$\beta_{1} -$$$$28\!\cdots\!60$$$$\beta_{2} +$$$$22\!\cdots\!47$$$$\beta_{3} +$$$$17\!\cdots\!48$$$$\beta_{4} -$$$$18\!\cdots\!31$$$$\beta_{5} -$$$$94\!\cdots\!54$$$$\beta_{6} +$$$$71\!\cdots\!11$$$$\beta_{7} -$$$$21\!\cdots\!44$$$$\beta_{8} +$$$$18\!\cdots\!19$$$$\beta_{9} -$$$$58\!\cdots\!11$$$$\beta_{10} +$$$$46\!\cdots\!01$$$$\beta_{11} -$$$$32\!\cdots\!29$$$$\beta_{12} -$$$$23\!\cdots\!12$$$$\beta_{13} -$$$$76\!\cdots\!76$$$$\beta_{14} -$$$$24\!\cdots\!32$$$$\beta_{15} -$$$$33\!\cdots\!87$$$$\beta_{16} -$$$$81\!\cdots\!28$$$$\beta_{17} -$$$$22\!\cdots\!24$$$$\beta_{18} -$$$$18\!\cdots\!20$$$$\beta_{19}) q^{81}$$ $$+($$$$48\!\cdots\!72$$$$-$$$$51\!\cdots\!50$$$$\beta_{1} +$$$$55\!\cdots\!18$$$$\beta_{2} -$$$$10\!\cdots\!02$$$$\beta_{3} -$$$$37\!\cdots\!42$$$$\beta_{4} +$$$$78\!\cdots\!98$$$$\beta_{5} +$$$$11\!\cdots\!44$$$$\beta_{6} +$$$$22\!\cdots\!96$$$$\beta_{7} -$$$$26\!\cdots\!94$$$$\beta_{8} +$$$$10\!\cdots\!48$$$$\beta_{9} +$$$$46\!\cdots\!42$$$$\beta_{10} +$$$$62\!\cdots\!04$$$$\beta_{11} +$$$$80\!\cdots\!48$$$$\beta_{12} -$$$$25\!\cdots\!96$$$$\beta_{13} -$$$$13\!\cdots\!76$$$$\beta_{15} -$$$$13\!\cdots\!40$$$$\beta_{16}) q^{82}$$ $$+($$$$15\!\cdots\!39$$$$+$$$$39\!\cdots\!26$$$$\beta_{1} -$$$$76\!\cdots\!57$$$$\beta_{2} -$$$$42\!\cdots\!05$$$$\beta_{3} -$$$$16\!\cdots\!20$$$$\beta_{4} -$$$$42\!\cdots\!37$$$$\beta_{5} +$$$$56\!\cdots\!26$$$$\beta_{6} +$$$$28\!\cdots\!80$$$$\beta_{7} -$$$$11\!\cdots\!29$$$$\beta_{8} +$$$$11\!\cdots\!02$$$$\beta_{9} +$$$$99\!\cdots\!61$$$$\beta_{10} +$$$$11\!\cdots\!55$$$$\beta_{11} -$$$$16\!\cdots\!02$$$$\beta_{12} -$$$$19\!\cdots\!40$$$$\beta_{13} -$$$$33\!\cdots\!09$$$$\beta_{14} -$$$$90\!\cdots\!58$$$$\beta_{15} -$$$$84\!\cdots\!12$$$$\beta_{16} -$$$$44\!\cdots\!92$$$$\beta_{17} +$$$$13\!\cdots\!24$$$$\beta_{18} +$$$$23\!\cdots\!16$$$$\beta_{19}) q^{83}$$ $$+(-$$$$14\!\cdots\!25$$$$-$$$$68\!\cdots\!28$$$$\beta_{1} -$$$$22\!\cdots\!57$$$$\beta_{2} +$$$$26\!\cdots\!68$$$$\beta_{3} -$$$$51\!\cdots\!70$$$$\beta_{4} -$$$$25\!\cdots\!07$$$$\beta_{5} -$$$$16\!\cdots\!27$$$$\beta_{6} +$$$$14\!\cdots\!41$$$$\beta_{7} +$$$$18\!\cdots\!03$$$$\beta_{8} -$$$$10\!\cdots\!27$$$$\beta_{9} -$$$$58\!\cdots\!47$$$$\beta_{10} +$$$$65\!\cdots\!76$$$$\beta_{11} +$$$$49\!\cdots\!47$$$$\beta_{12} +$$$$80\!\cdots\!35$$$$\beta_{13} +$$$$19\!\cdots\!42$$$$\beta_{14} +$$$$24\!\cdots\!46$$$$\beta_{15} +$$$$12\!\cdots\!97$$$$\beta_{16} +$$$$96\!\cdots\!22$$$$\beta_{17} +$$$$14\!\cdots\!96$$$$\beta_{18} +$$$$10\!\cdots\!40$$$$\beta_{19}) q^{84}$$ $$+($$$$78\!\cdots\!00$$$$-$$$$39\!\cdots\!84$$$$\beta_{1} +$$$$42\!\cdots\!04$$$$\beta_{2} -$$$$24\!\cdots\!24$$$$\beta_{3} -$$$$39\!\cdots\!80$$$$\beta_{4} -$$$$30\!\cdots\!00$$$$\beta_{5} +$$$$83\!\cdots\!84$$$$\beta_{6} +$$$$15\!\cdots\!84$$$$\beta_{7} +$$$$72\!\cdots\!84$$$$\beta_{8} -$$$$37\!\cdots\!48$$$$\beta_{9} +$$$$26\!\cdots\!44$$$$\beta_{10} +$$$$26\!\cdots\!24$$$$\beta_{11} -$$$$13\!\cdots\!52$$$$\beta_{12} +$$$$31\!\cdots\!04$$$$\beta_{13} +$$$$23\!\cdots\!36$$$$\beta_{15} +$$$$43\!\cdots\!92$$$$\beta_{16}) q^{85}$$ $$+($$$$15\!\cdots\!75$$$$+$$$$13\!\cdots\!28$$$$\beta_{1} -$$$$79\!\cdots\!98$$$$\beta_{2} -$$$$43\!\cdots\!57$$$$\beta_{3} +$$$$24\!\cdots\!91$$$$\beta_{4} -$$$$27\!\cdots\!75$$$$\beta_{5} -$$$$32\!\cdots\!58$$$$\beta_{6} +$$$$33\!\cdots\!22$$$$\beta_{7} -$$$$84\!\cdots\!69$$$$\beta_{8} +$$$$48\!\cdots\!15$$$$\beta_{9} -$$$$89\!\cdots\!21$$$$\beta_{10} -$$$$85\!\cdots\!21$$$$\beta_{11} +$$$$54\!\cdots\!70$$$$\beta_{12} +$$$$85\!\cdots\!50$$$$\beta_{13} +$$$$31\!\cdots\!36$$$$\beta_{14} +$$$$74\!\cdots\!90$$$$\beta_{15} +$$$$57\!\cdots\!60$$$$\beta_{16} +$$$$17\!\cdots\!80$$$$\beta_{17} +$$$$13\!\cdots\!30$$$$\beta_{18} +$$$$50\!\cdots\!20$$$$\beta_{19}) q^{86}$$ $$+(-$$$$10\!\cdots\!59$$$$+$$$$35\!\cdots\!92$$$$\beta_{1} -$$$$59\!\cdots\!13$$$$\beta_{2} +$$$$17\!\cdots\!21$$$$\beta_{3} -$$$$28\!\cdots\!10$$$$\beta_{4} -$$$$50\!\cdots\!39$$$$\beta_{5} +$$$$18\!\cdots\!92$$$$\beta_{6} +$$$$32\!\cdots\!31$$$$\beta_{7} +$$$$58\!\cdots\!46$$$$\beta_{8} +$$$$49\!\cdots\!55$$$$\beta_{9} +$$$$21\!\cdots\!93$$$$\beta_{10} -$$$$31\!\cdots\!05$$$$\beta_{11} -$$$$90\!\cdots\!05$$$$\beta_{12} -$$$$16\!\cdots\!17$$$$\beta_{13} +$$$$88\!\cdots\!69$$$$\beta_{14} +$$$$16\!\cdots\!86$$$$\beta_{15} -$$$$17\!\cdots\!99$$$$\beta_{16} +$$$$58\!\cdots\!95$$$$\beta_{17} -$$$$49\!\cdots\!07$$$$\beta_{18} -$$$$27\!\cdots\!03$$$$\beta_{19}) q^{87}$$ $$+($$$$21\!\cdots\!30$$$$-$$$$26\!\cdots\!00$$$$\beta_{1} +$$$$28\!\cdots\!60$$$$\beta_{2} -$$$$61\!\cdots\!48$$$$\beta_{3} -$$$$27\!\cdots\!90$$$$\beta_{4} -$$$$18\!\cdots\!20$$$$\beta_{5} -$$$$16\!\cdots\!02$$$$\beta_{6} +$$$$93\!\cdots\!06$$$$\beta_{7} +$$$$36\!\cdots\!32$$$$\beta_{8} -$$$$84\!\cdots\!54$$$$\beta_{9} -$$$$56\!\cdots\!64$$$$\beta_{10} -$$$$10\!\cdots\!96$$$$\beta_{11} +$$$$72\!\cdots\!16$$$$\beta_{12} +$$$$14\!\cdots\!54$$$$\beta_{13} +$$$$37\!\cdots\!38$$$$\beta_{15} -$$$$23\!\cdots\!82$$$$\beta_{16}) q^{88}$$ $$+($$$$26\!\cdots\!13$$$$+$$$$53\!\cdots\!20$$$$\beta_{1} -$$$$13\!\cdots\!08$$$$\beta_{2} -$$$$72\!\cdots\!51$$$$\beta_{3} +$$$$25\!\cdots\!04$$$$\beta_{4} +$$$$13\!\cdots\!83$$$$\beta_{5} -$$$$41\!\cdots\!22$$$$\beta_{6} +$$$$67\!\cdots\!57$$$$\beta_{7} +$$$$17\!\cdots\!00$$$$\beta_{8} +$$$$61\!\cdots\!89$$$$\beta_{9} -$$$$12\!\cdots\!73$$$$\beta_{10} -$$$$12\!\cdots\!65$$$$\beta_{11} -$$$$13\!\cdots\!79$$$$\beta_{12} -$$$$14\!\cdots\!16$$$$\beta_{13} -$$$$19\!\cdots\!28$$$$\beta_{14} +$$$$30\!\cdots\!64$$$$\beta_{15} -$$$$53\!\cdots\!85$$$$\beta_{16} -$$$$34\!\cdots\!24$$$$\beta_{17} -$$$$38\!\cdots\!72$$$$\beta_{18} -$$$$26\!\cdots\!80$$$$\beta_{19}) q^{89}$$ $$+(-$$$$34\!\cdots\!70$$$$+$$$$53\!\cdots\!11$$$$\beta_{1} +$$$$67\!\cdots\!53$$$$\beta_{2} +$$$$77\!\cdots\!53$$$$\beta_{3} -$$$$92\!\cdots\!93$$$$\beta_{4} +$$$$44\!\cdots\!65$$$$\beta_{5} +$$$$98\!\cdots\!56$$$$\beta_{6} +$$$$14\!\cdots\!24$$$$\beta_{7} -$$$$12\!\cdots\!09$$$$\beta_{8} -$$$$80\!\cdots\!42$$$$\beta_{9} -$$$$13\!\cdots\!71$$$$\beta_{10} -$$$$21\!\cdots\!44$$$$\beta_{11} +$$$$12\!\cdots\!32$$$$\beta_{12} +$$$$24\!\cdots\!26$$$$\beta_{13} -$$$$59\!\cdots\!40$$$$\beta_{14} -$$$$32\!\cdots\!76$$$$\beta_{15} -$$$$18\!\cdots\!52$$$$\beta_{16} +$$$$90\!\cdots\!80$$$$\beta_{17} +$$$$71\!\cdots\!10$$$$\beta_{18} +$$$$32\!\cdots\!60$$$$\beta_{19}) q^{90}$$ $$+(-$$$$83\!\cdots\!34$$$$-$$$$17\!\cdots\!98$$$$\beta_{1} +$$$$19\!\cdots\!01$$$$\beta_{2} -$$$$17\!\cdots\!90$$$$\beta_{3} -$$$$16\!\cdots\!77$$$$\beta_{4} -$$$$88\!\cdots\!33$$$$\beta_{5} +$$$$27\!\cdots\!88$$$$\beta_{6} +$$$$57\!\cdots\!74$$$$\beta_{7} +$$$$13\!\cdots\!27$$$$\beta_{8} +$$$$11\!\cdots\!96$$$$\beta_{9} +$$$$33\!\cdots\!97$$$$\beta_{10} -$$$$71\!\cdots\!80$$$$\beta_{11} -$$$$15\!\cdots\!77$$$$\beta_{12} -$$$$78\!\cdots\!54$$$$\beta_{13} -$$$$12\!\cdots\!41$$$$\beta_{15} +$$$$91\!\cdots\!12$$$$\beta_{16}) q^{91}$$ $$+($$$$15\!\cdots\!38$$$$-$$$$18\!\cdots\!72$$$$\beta_{1} -$$$$79\!\cdots\!74$$$$\beta_{2} -$$$$43\!\cdots\!16$$$$\beta_{3} +$$$$59\!\cdots\!80$$$$\beta_{4} -$$$$67\!\cdots\!30$$$$\beta_{5} -$$$$15\!\cdots\!30$$$$\beta_{6} +$$$$32\!\cdots\!02$$$$\beta_{7} +$$$$39\!\cdots\!98$$$$\beta_{8} -$$$$68\!\cdots\!90$$$$\beta_{9} -$$$$26\!\cdots\!06$$$$\beta_{10} -$$$$39\!\cdots\!72$$$$\beta_{11} +$$$$25\!\cdots\!38$$$$\beta_{12} +$$$$44\!\cdots\!46$$$$\beta_{13} +$$$$74\!\cdots\!12$$$$\beta_{14} -$$$$82\!\cdots\!56$$$$\beta_{15} -$$$$22\!\cdots\!82$$$$\beta_{16} +$$$$33\!\cdots\!44$$$$\beta_{17} +$$$$21\!\cdots\!20$$$$\beta_{18} +$$$$69\!\cdots\!00$$$$\beta_{19}) q^{92}$$ $$+($$$$25\!\cdots\!30$$$$+$$$$60\!\cdots\!81$$$$\beta_{1} -$$$$25\!\cdots\!93$$$$\beta_{2} -$$$$31\!\cdots\!06$$$$\beta_{3} -$$$$15\!\cdots\!06$$$$\beta_{4} +$$$$24\!\cdots\!93$$$$\beta_{5} +$$$$28\!\cdots\!64$$$$\beta_{6} +$$$$22\!\cdots\!57$$$$\beta_{7} -$$$$14\!\cdots\!10$$$$\beta_{8} -$$$$89\!\cdots\!48$$$$\beta_{9} -$$$$88\!\cdots\!47$$$$\beta_{10} +$$$$15\!\cdots\!05$$$$\beta_{11} -$$$$21\!\cdots\!40$$$$\beta_{12} -$$$$45\!\cdots\!13$$$$\beta_{13} +$$$$11\!\cdots\!24$$$$\beta_{14} -$$$$16\!\cdots\!57$$$$\beta_{15} +$$$$36\!\cdots\!74$$$$\beta_{16} -$$$$97\!\cdots\!56$$$$\beta_{17} +$$$$22\!\cdots\!84$$$$\beta_{18} +$$$$53\!\cdots\!76$$$$\beta_{19}) q^{93}$$ $$+(-$$$$12\!\cdots\!84$$$$-$$$$14\!\cdots\!40$$$$\beta_{1} +$$$$15\!\cdots\!84$$$$\beta_{2} +$$$$89\!\cdots\!16$$$$\beta_{3} -$$$$16\!\cdots\!88$$$$\beta_{4} +$$$$66\!\cdots\!04$$$$\beta_{5} -$$$$21\!\cdots\!20$$$$\beta_{6} +$$$$67\!\cdots\!80$$$$\beta_{7} -$$$$13\!\cdots\!32$$$$\beta_{8} +$$$$67\!\cdots\!56$$$$\beta_{9} +$$$$44\!\cdots\!12$$$$\beta_{10} +$$$$24\!\cdots\!40$$$$\beta_{11} -$$$$64\!\cdots\!48$$$$\beta_{12} +$$$$17\!\cdots\!00$$$$\beta_{13} -$$$$30\!\cdots\!40$$$$\beta_{15} -$$$$24\!\cdots\!16$$$$\beta_{16}) q^{94}$$ $$+($$$$33\!\cdots\!34$$$$+$$$$24\!\cdots\!50$$$$\beta_{1} -$$$$16\!\cdots\!41$$$$\beta_{2} -$$$$94\!\cdots\!46$$$$\beta_{3} -$$$$51\!\cdots\!87$$$$\beta_{4} -$$$$86\!\cdots\!07$$$$\beta_{5} +$$$$13\!\cdots\!00$$$$\beta_{6} +$$$$62\!\cdots\!24$$$$\beta_{7} -$$$$77\!\cdots\!41$$$$\beta_{8} -$$$$12\!\cdots\!86$$$$\beta_{9} +$$$$33\!\cdots\!35$$$$\beta_{10} +$$$$33\!\cdots\!36$$$$\beta_{11} -$$$$34\!\cdots\!37$$$$\beta_{12} -$$$$34\!\cdots\!24$$$$\beta_{13} -$$$$44\!\cdots\!12$$$$\beta_{14} +$$$$15\!\cdots\!61$$$$\beta_{15} -$$$$98\!\cdots\!22$$$$\beta_{16} -$$$$84\!\cdots\!26$$$$\beta_{17} +$$$$48\!\cdots\!78$$$$\beta_{18} -$$$$61\!\cdots\!02$$$$\beta_{19}) q^{95}$$ $$+($$$$85\!\cdots\!08$$$$-$$$$26\!\cdots\!04$$$$\beta_{1} +$$$$51\!\cdots\!12$$$$\beta_{2} -$$$$20\!\cdots\!16$$$$\beta_{3} +$$$$13\!\cdots\!48$$$$\beta_{4} -$$$$24\!\cdots\!96$$$$\beta_{5} -$$$$19\!\cdots\!80$$$$\beta_{6} +$$$$14\!\cdots\!12$$$$\beta_{7} +$$$$99\!\cdots\!20$$$$\beta_{8} +$$$$69\!\cdots\!16$$$$\beta_{9} -$$$$43\!\cdots\!32$$$$\beta_{10} +$$$$15\!\cdots\!72$$$$\beta_{11} +$$$$10\!\cdots\!56$$$$\beta_{12} +$$$$71\!\cdots\!60$$$$\beta_{13} +$$$$23\!\cdots\!12$$$$\beta_{14} +$$$$74\!\cdots\!48$$$$\beta_{15} -$$$$30\!\cdots\!24$$$$\beta_{16} +$$$$36\!\cdots\!36$$$$\beta_{17} -$$$$16\!\cdots\!52$$$$\beta_{18} -$$$$28\!\cdots\!40$$$$\beta_{19}) q^{96}$$ $$+(-$$$$10\!\cdots\!13$$$$-$$$$25\!\cdots\!26$$$$\beta_{1} +$$$$27\!\cdots\!86$$$$\beta_{2} +$$$$15\!\cdots\!31$$$$\beta_{3} -$$$$29\!\cdots\!74$$$$\beta_{4} +$$$$15\!\cdots\!11$$$$\beta_{5} +$$$$84\!\cdots\!10$$$$\beta_{6} +$$$$13\!\cdots\!27$$$$\beta_{7} -$$$$44\!\cdots\!02$$$$\beta_{8} -$$$$20\!\cdots\!11$$$$\beta_{9} -$$$$10\!\cdots\!83$$$$\beta_{10} +$$$$19\!\cdots\!05$$$$\beta_{11} +$$$$10\!\cdots\!79$$$$\beta_{12} -$$$$22\!\cdots\!20$$$$\beta_{13} +$$$$60\!\cdots\!12$$$$\beta_{15} +$$$$29\!\cdots\!49$$$$\beta_{16}) q^{97}$$ $$+($$$$98\!\cdots\!50$$$$+$$$$12\!\cdots\!59$$$$\beta_{1} -$$$$49\!\cdots\!20$$$$\beta_{2} -$$$$27\!\cdots\!84$$$$\beta_{3} -$$$$12\!\cdots\!58$$$$\beta_{4} -$$$$60\!\cdots\!78$$$$\beta_{5} +$$$$13\!\cdots\!66$$$$\beta_{6} +$$$$15\!\cdots\!82$$$$\beta_{7} -$$$$18\!\cdots\!00$$$$\beta_{8} +$$$$65\!\cdots\!62$$$$\beta_{9} +$$$$39\!\cdots\!04$$$$\beta_{10} +$$$$40\!\cdots\!96$$$$\beta_{11} +$$$$12\!\cdots\!34$$$$\beta_{12} +$$$$36\!\cdots\!42$$$$\beta_{13} +$$$$11\!\cdots\!40$$$$\beta_{14} +$$$$73\!\cdots\!10$$$$\beta_{15} +$$$$37\!\cdots\!04$$$$\beta_{16} +$$$$62\!\cdots\!76$$$$\beta_{17} +$$$$56\!\cdots\!14$$$$\beta_{18} -$$$$31\!\cdots\!44$$$$\beta_{19}) q^{98}$$ $$+($$$$65\!\cdots\!98$$$$-$$$$39\!\cdots\!77$$$$\beta_{1} -$$$$35\!\cdots\!32$$$$\beta_{2} -$$$$13\!\cdots\!71$$$$\beta_{3} +$$$$76\!\cdots\!13$$$$\beta_{4} +$$$$38\!\cdots\!31$$$$\beta_{5} +$$$$20\!\cdots\!38$$$$\beta_{6} +$$$$13\!\cdots\!57$$$$\beta_{7} +$$$$17\!\cdots\!31$$$$\beta_{8} +$$$$23\!\cdots\!46$$$$\beta_{9} +$$$$41\!\cdots\!11$$$$\beta_{10} +$$$$19\!\cdots\!30$$$$\beta_{11} -$$$$22\!\cdots\!95$$$$\beta_{12} +$$$$15\!\cdots\!24$$$$\beta_{13} -$$$$91\!\cdots\!78$$$$\beta_{14} -$$$$98\!\cdots\!41$$$$\beta_{15} +$$$$11\!\cdots\!38$$$$\beta_{16} -$$$$55\!\cdots\!74$$$$\beta_{17} +$$$$47\!\cdots\!58$$$$\beta_{18} +$$$$30\!\cdots\!50$$$$\beta_{19}) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q$$ $$\mathstrut +\mathstrut 4870975596954516q^{3}$$ $$\mathstrut -\mathstrut$$$$87\!\cdots\!80$$$$q^{4}$$ $$\mathstrut -\mathstrut$$$$36\!\cdots\!80$$$$q^{6}$$ $$\mathstrut +\mathstrut$$$$12\!\cdots\!88$$$$q^{7}$$ $$\mathstrut +\mathstrut$$$$15\!\cdots\!00$$$$q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$20q$$ $$\mathstrut +\mathstrut 4870975596954516q^{3}$$ $$\mathstrut -\mathstrut$$$$87\!\cdots\!80$$$$q^{4}$$ $$\mathstrut -\mathstrut$$$$36\!\cdots\!80$$$$q^{6}$$ $$\mathstrut +\mathstrut$$$$12\!\cdots\!88$$$$q^{7}$$ $$\mathstrut +\mathstrut$$$$15\!\cdots\!00$$$$q^{9}$$ $$\mathstrut -\mathstrut$$$$78\!\cdots\!00$$$$q^{10}$$ $$\mathstrut +\mathstrut$$$$72\!\cdots\!64$$$$q^{12}$$ $$\mathstrut +\mathstrut$$$$18\!\cdots\!12$$$$q^{13}$$ $$\mathstrut +\mathstrut$$$$29\!\cdots\!00$$$$q^{15}$$ $$\mathstrut +\mathstrut$$$$35\!\cdots\!60$$$$q^{16}$$ $$\mathstrut -\mathstrut$$$$13\!\cdots\!20$$$$q^{18}$$ $$\mathstrut -\mathstrut$$$$21\!\cdots\!00$$$$q^{19}$$ $$\mathstrut +\mathstrut$$$$84\!\cdots\!20$$$$q^{21}$$ $$\mathstrut -\mathstrut$$$$19\!\cdots\!20$$$$q^{22}$$ $$\mathstrut -\mathstrut$$$$85\!\cdots\!60$$$$q^{24}$$ $$\mathstrut -\mathstrut$$$$30\!\cdots\!00$$$$q^{25}$$ $$\mathstrut +\mathstrut$$$$55\!\cdots\!64$$$$q^{27}$$ $$\mathstrut +\mathstrut$$$$14\!\cdots\!92$$$$q^{28}$$ $$\mathstrut -\mathstrut$$$$13\!\cdots\!00$$$$q^{30}$$ $$\mathstrut +\mathstrut$$$$61\!\cdots\!40$$$$q^{31}$$ $$\mathstrut +\mathstrut$$$$76\!\cdots\!60$$$$q^{33}$$ $$\mathstrut -\mathstrut$$$$60\!\cdots\!60$$$$q^{34}$$ $$\mathstrut -\mathstrut$$$$31\!\cdots\!60$$$$q^{36}$$ $$\mathstrut +\mathstrut$$$$27\!\cdots\!28$$$$q^{37}$$ $$\mathstrut +\mathstrut$$$$55\!\cdots\!80$$$$q^{39}$$ $$\mathstrut -\mathstrut$$$$16\!\cdots\!00$$$$q^{40}$$ $$\mathstrut -\mathstrut$$$$36\!\cdots\!60$$$$q^{42}$$ $$\mathstrut +\mathstrut$$$$29\!\cdots\!32$$$$q^{43}$$ $$\mathstrut +\mathstrut$$$$28\!\cdots\!00$$$$q^{45}$$ $$\mathstrut -\mathstrut$$$$93\!\cdots\!60$$$$q^{46}$$ $$\mathstrut +\mathstrut$$$$44\!\cdots\!96$$$$q^{48}$$ $$\mathstrut +\mathstrut$$$$35\!\cdots\!80$$$$q^{49}$$ $$\mathstrut +\mathstrut$$$$24\!\cdots\!20$$$$q^{51}$$ $$\mathstrut -\mathstrut$$$$70\!\cdots\!12$$$$q^{52}$$ $$\mathstrut -\mathstrut$$$$38\!\cdots\!80$$$$q^{54}$$ $$\mathstrut +\mathstrut$$$$13\!\cdots\!00$$$$q^{55}$$ $$\mathstrut +\mathstrut$$$$14\!\cdots\!68$$$$q^{57}$$ $$\mathstrut -\mathstrut$$$$10\!\cdots\!20$$$$q^{58}$$ $$\mathstrut +\mathstrut$$$$18\!\cdots\!00$$$$q^{60}$$ $$\mathstrut +\mathstrut$$$$36\!\cdots\!40$$$$q^{61}$$ $$\mathstrut +\mathstrut$$$$13\!\cdots\!72$$$$q^{63}$$ $$\mathstrut -\mathstrut$$$$42\!\cdots\!40$$$$q^{64}$$ $$\mathstrut +\mathstrut$$$$83\!\cdots\!00$$$$q^{66}$$ $$\mathstrut -\mathstrut$$$$62\!\cdots\!32$$$$q^{67}$$ $$\mathstrut +\mathstrut$$$$26\!\cdots\!20$$$$q^{69}$$ $$\mathstrut -\mathstrut$$$$53\!\cdots\!00$$$$q^{70}$$ $$\mathstrut +\mathstrut$$$$13\!\cdots\!20$$$$q^{72}$$ $$\mathstrut -\mathstrut$$$$23\!\cdots\!28$$$$q^{73}$$ $$\mathstrut +\mathstrut$$$$12\!\cdots\!00$$$$q^{75}$$ $$\mathstrut -\mathstrut$$$$88\!\cdots\!20$$$$q^{76}$$ $$\mathstrut +\mathstrut$$$$42\!\cdots\!00$$$$q^{78}$$ $$\mathstrut -\mathstrut$$$$18\!\cdots\!00$$$$q^{79}$$ $$\mathstrut -\mathstrut$$$$26\!\cdots\!80$$$$q^{81}$$ $$\mathstrut +\mathstrut$$$$97\!\cdots\!20$$$$q^{82}$$ $$\mathstrut -\mathstrut$$$$29\!\cdots\!20$$$$q^{84}$$ $$\mathstrut +\mathstrut$$$$15\!\cdots\!00$$$$q^{85}$$ $$\mathstrut -\mathstrut$$$$20\!\cdots\!00$$$$q^{87}$$ $$\mathstrut +\mathstrut$$$$43\!\cdots\!80$$$$q^{88}$$ $$\mathstrut -\mathstrut$$$$68\!\cdots\!00$$$$q^{90}$$ $$\mathstrut -\mathstrut$$$$16\!\cdots\!60$$$$q^{91}$$ $$\mathstrut +\mathstrut$$$$50\!\cdots\!32$$$$q^{93}$$ $$\mathstrut -\mathstrut$$$$24\!\cdots\!60$$$$q^{94}$$ $$\mathstrut +\mathstrut$$$$17\!\cdots\!60$$$$q^{96}$$ $$\mathstrut -\mathstrut$$$$20\!\cdots\!52$$$$q^{97}$$ $$\mathstrut +\mathstrut$$$$13\!\cdots\!00$$$$q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20}\mathstrut +\mathstrut$$ $$906717684887249855$$ $$x^{18}\mathstrut +\mathstrut$$ $$34\!\cdots\!60$$ $$x^{16}\mathstrut +\mathstrut$$ $$71\!\cdots\!60$$ $$x^{14}\mathstrut +\mathstrut$$ $$89\!\cdots\!80$$ $$x^{12}\mathstrut +\mathstrut$$ $$68\!\cdots\!76$$ $$x^{10}\mathstrut +\mathstrut$$ $$31\!\cdots\!00$$ $$x^{8}\mathstrut +\mathstrut$$ $$84\!\cdots\!00$$ $$x^{6}\mathstrut +\mathstrut$$ $$11\!\cdots\!00$$ $$x^{4}\mathstrut +\mathstrut$$ $$60\!\cdots\!00$$ $$x^{2}\mathstrut +\mathstrut$$ $$56\!\cdots\!00$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$36 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$70\!\cdots\!75$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$30\!\cdots\!44$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$64\!\cdots\!25$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$27\!\cdots\!20$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$24\!\cdots\!00$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$10\!\cdots\!80$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$50\!\cdots\!00$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$21\!\cdots\!00$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$63\!\cdots\!00$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$26\!\cdots\!60$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$48\!\cdots\!00$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$20\!\cdots\!04$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$22\!\cdots\!00$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$95\!\cdots\!40$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$59\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$24\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$81\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$24\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$42\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$11\!\cdots\!00$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$70\!\cdots\!75$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$30\!\cdots\!44$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$64\!\cdots\!25$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$27\!\cdots\!20$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$24\!\cdots\!00$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$10\!\cdots\!80$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$50\!\cdots\!00$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$21\!\cdots\!00$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$63\!\cdots\!00$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$26\!\cdots\!60$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$48\!\cdots\!00$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$20\!\cdots\!04$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$22\!\cdots\!00$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$95\!\cdots\!40$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$59\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$24\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$81\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$31\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$42\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$59\!\cdots\!00$$$$)/$$$$49\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$44\!\cdots\!05$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$74\!\cdots\!52$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$40\!\cdots\!75$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$67\!\cdots\!60$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$15\!\cdots\!00$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$25\!\cdots\!40$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$31\!\cdots\!00$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$52\!\cdots\!00$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$39\!\cdots\!00$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$65\!\cdots\!80$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$30\!\cdots\!80$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$50\!\cdots\!32$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$14\!\cdots\!00$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$23\!\cdots\!20$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$38\!\cdots\!00$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$61\!\cdots\!00$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$52\!\cdots\!00$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$62\!\cdots\!00$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$28\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$42\!\cdots\!00$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$47\!\cdots\!59$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$39\!\cdots\!52$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$42\!\cdots\!45$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$30\!\cdots\!60$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$16\!\cdots\!80$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$11\!\cdots\!40$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$33\!\cdots\!00$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$33\!\cdots\!00$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$42\!\cdots\!60$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$64\!\cdots\!80$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$32\!\cdots\!44$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$75\!\cdots\!32$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$15\!\cdots\!40$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$48\!\cdots\!20$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$40\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$14\!\cdots\!00$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$55\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$12\!\cdots\!00$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$29\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$54\!\cdots\!00$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$33\!\cdots\!31$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$49\!\cdots\!48$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$30\!\cdots\!05$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$45\!\cdots\!20$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$11\!\cdots\!20$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$17\!\cdots\!80$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$23\!\cdots\!00$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$37\!\cdots\!80$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$29\!\cdots\!40$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$46\!\cdots\!40$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$22\!\cdots\!96$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$34\!\cdots\!48$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$10\!\cdots\!60$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$14\!\cdots\!80$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$28\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$33\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$38\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$26\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$20\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$80\!\cdots\!00$$$$)/$$$$68\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$93\!\cdots\!67$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$38\!\cdots\!20$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$84\!\cdots\!85$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$34\!\cdots\!00$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$32\!\cdots\!40$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$13\!\cdots\!00$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$67\!\cdots\!00$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$27\!\cdots\!00$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$83\!\cdots\!80$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$34\!\cdots\!00$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$63\!\cdots\!72$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$27\!\cdots\!20$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$29\!\cdots\!20$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$13\!\cdots\!00$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$79\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$34\!\cdots\!00$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$10\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$34\!\cdots\!00$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$56\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$19\!\cdots\!00$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!11$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$26\!\cdots\!16$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$17\!\cdots\!05$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$21\!\cdots\!80$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$66\!\cdots\!20$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$84\!\cdots\!20$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$13\!\cdots\!00$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$20\!\cdots\!00$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$17\!\cdots\!40$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$33\!\cdots\!40$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$13\!\cdots\!76$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$34\!\cdots\!56$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$61\!\cdots\!60$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$20\!\cdots\!60$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$16\!\cdots\!00$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$59\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$22\!\cdots\!00$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$55\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$11\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$11\!\cdots\!00$$$$)/$$$$47\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$55\!\cdots\!33$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$16\!\cdots\!60$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$50\!\cdots\!55$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$14\!\cdots\!80$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$19\!\cdots\!20$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$52\!\cdots\!20$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$40\!\cdots\!40$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$10\!\cdots\!80$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$50\!\cdots\!80$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$11\!\cdots\!20$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$39\!\cdots\!68$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$80\!\cdots\!40$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$18\!\cdots\!80$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$31\!\cdots\!00$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$50\!\cdots\!00$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$62\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$69\!\cdots\!00$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$47\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$37\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$96\!\cdots\!00$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$37\!\cdots\!99$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$26\!\cdots\!72$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$33\!\cdots\!45$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$17\!\cdots\!40$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$12\!\cdots\!80$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$26\!\cdots\!60$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$26\!\cdots\!00$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$10\!\cdots\!00$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$33\!\cdots\!60$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$19\!\cdots\!20$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$25\!\cdots\!84$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$19\!\cdots\!48$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$11\!\cdots\!40$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$11\!\cdots\!80$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$31\!\cdots\!00$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$29\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$42\!\cdots\!00$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$26\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$22\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$70\!\cdots\!00$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$84\!\cdots\!43$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$35\!\cdots\!24$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$76\!\cdots\!25$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$21\!\cdots\!40$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$29\!\cdots\!00$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$18\!\cdots\!60$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$60\!\cdots\!60$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$15\!\cdots\!80$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$75\!\cdots\!60$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$52\!\cdots\!60$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$57\!\cdots\!48$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$71\!\cdots\!96$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$26\!\cdots\!80$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$47\!\cdots\!60$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$71\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$14\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$97\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$13\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$51\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$83\!\cdots\!00$$$$)/$$$$25\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$34\!\cdots\!15$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$35\!\cdots\!12$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$31\!\cdots\!65$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$29\!\cdots\!60$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$12\!\cdots\!60$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$99\!\cdots\!40$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$25\!\cdots\!40$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$18\!\cdots\!00$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$31\!\cdots\!60$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$18\!\cdots\!80$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$24\!\cdots\!80$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$11\!\cdots\!92$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$11\!\cdots\!00$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$38\!\cdots\!20$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$31\!\cdots\!00$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$66\!\cdots\!00$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$43\!\cdots\!00$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$45\!\cdots\!00$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$23\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$57\!\cdots\!00$$$$)/$$$$25\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$55\!\cdots\!03$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$84\!\cdots\!64$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$50\!\cdots\!65$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$74\!\cdots\!60$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$19\!\cdots\!60$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$27\!\cdots\!40$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$40\!\cdots\!00$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$54\!\cdots\!40$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$51\!\cdots\!20$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$62\!\cdots\!20$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$39\!\cdots\!48$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$43\!\cdots\!64$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$18\!\cdots\!80$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$17\!\cdots\!40$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$50\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$34\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$70\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$25\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$37\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$85\!\cdots\!00$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$28\!\cdots\!69$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$26\!\cdots\!48$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$26\!\cdots\!35$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$23\!\cdots\!20$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$99\!\cdots\!40$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$83\!\cdots\!80$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$20\!\cdots\!40$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$16\!\cdots\!80$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$25\!\cdots\!20$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$18\!\cdots\!40$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$19\!\cdots\!44$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$12\!\cdots\!48$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$91\!\cdots\!40$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$48\!\cdots\!80$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$24\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$96\!\cdots\!00$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$33\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$72\!\cdots\!00$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$17\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$15\!\cdots\!00$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!87$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$12\!\cdots\!16$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$18\!\cdots\!85$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$10\!\cdots\!40$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$70\!\cdots\!40$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$38\!\cdots\!60$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$14\!\cdots\!00$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$73\!\cdots\!60$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$18\!\cdots\!80$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$82\!\cdots\!80$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$14\!\cdots\!92$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$54\!\cdots\!16$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$67\!\cdots\!20$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$20\!\cdots\!60$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$18\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$40\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$25\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$30\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$13\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$78\!\cdots\!00$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$42\!\cdots\!01$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$52\!\cdots\!64$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$39\!\cdots\!95$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$39\!\cdots\!80$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$15\!\cdots\!80$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$11\!\cdots\!20$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$31\!\cdots\!40$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$14\!\cdots\!60$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$39\!\cdots\!00$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$79\!\cdots\!00$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$30\!\cdots\!56$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$17\!\cdots\!16$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$14\!\cdots\!60$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$37\!\cdots\!60$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$39\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$14\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$54\!\cdots\!00$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$16\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$29\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$13\!\cdots\!00$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$99\!\cdots\!49$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$14\!\cdots\!32$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$89\!\cdots\!75$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$13\!\cdots\!00$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$34\!\cdots\!00$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$50\!\cdots\!00$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$70\!\cdots\!80$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$10\!\cdots\!40$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$88\!\cdots\!80$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$12\!\cdots\!40$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$67\!\cdots\!64$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$88\!\cdots\!52$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$31\!\cdots\!40$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$36\!\cdots\!20$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$83\!\cdots\!00$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$76\!\cdots\!00$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$11\!\cdots\!00$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$57\!\cdots\!00$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$59\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$12\!\cdots\!00$$$$)/$$$$84\!\cdots\!00$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$16\!\cdots\!53$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$10\!\cdots\!12$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$15\!\cdots\!15$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$82\!\cdots\!20$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$60\!\cdots\!60$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$26\!\cdots\!80$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$12\!\cdots\!00$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$43\!\cdots\!60$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$15\!\cdots\!20$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$38\!\cdots\!20$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$10\!\cdots\!48$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$16\!\cdots\!52$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$42\!\cdots\!80$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$22\!\cdots\!20$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$65\!\cdots\!00$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$57\!\cdots\!00$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$33\!\cdots\!00$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$10\!\cdots\!00$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$10\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$53\!\cdots\!00$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{19}$$ $$=$$ $$($$$$25\!\cdots\!29$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$15\!\cdots\!24$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$22\!\cdots\!15$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$13\!\cdots\!00$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$86\!\cdots\!60$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$49\!\cdots\!00$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$17\!\cdots\!20$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$95\!\cdots\!80$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$21\!\cdots\!40$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$10\!\cdots\!80$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$16\!\cdots\!84$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$73\!\cdots\!64$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$73\!\cdots\!40$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$28\!\cdots\!40$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$18\!\cdots\!00$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$56\!\cdots\!00$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$24\!\cdots\!00$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$40\!\cdots\!00$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$11\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$11\!\cdots\!00$$$$)/$$$$63\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/36$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3}\mathstrut -\mathstrut$$ $$1519$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$141$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$117510611961387580904$$$$)/1296$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{8}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$227$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$671787$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$90503341$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$16361058157166$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$189993356209881425732$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3272193665213$$$$)/46656$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$13575$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$29517$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$17257$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$26976$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$227556$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1385528$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$2058155$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$81866204$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$84058466325$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$162598667649$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$27337238790086$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$144494037007035$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$281198672344971877176$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$278605710987654183234672$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$25814672224335600045120$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$22326235735894135855925494559722770113807$$$$)/1679616$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$2575287648$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$13548022232$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$5827627322860$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$1453260582654$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$7224069366228$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$50420438188996$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$1659694146574$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$3294119686926$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$17806451342152$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$30116245912354$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$63930164527513854$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$22190540523448070521$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$203549443232772264335$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$472855645559796338$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$828873915251826129931$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$17319007876090264910699997$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2687385612605780775380247555$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$485695741189351620476081801387248$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2665569763599380936657250234195015681232$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$97138614224876300456438161121757$$$$)/3779136$$ $$\nu^{6}$$ $$=$$ $$($$$$39\!\cdots\!73$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$71\!\cdots\!75$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$40\!\cdots\!15$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$65\!\cdots\!96$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$76\!\cdots\!28$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$43\!\cdots\!52$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$31\!\cdots\!01$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$25\!\cdots\!84$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$19\!\cdots\!11$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$41\!\cdots\!55$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$93\!\cdots\!26$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$26\!\cdots\!93$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$47\!\cdots\!60$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$43\!\cdots\!40$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$40\!\cdots\!60$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$31\!\cdots\!01$$$$)/$$$$136048896$$ $$\nu^{7}$$ $$=$$ $$($$$$73\!\cdots\!60$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$35\!\cdots\!40$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$18\!\cdots\!84$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$37\!\cdots\!02$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$20\!\cdots\!96$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$12\!\cdots\!12$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$55\!\cdots\!86$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$10\!\cdots\!78$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$11\!\cdots\!28$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$14\!\cdots\!74$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$18\!\cdots\!50$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$41\!\cdots\!39$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$47\!\cdots\!73$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$35\!\cdots\!50$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$13\!\cdots\!47$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$39\!\cdots\!39$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$64\!\cdots\!85$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$11\!\cdots\!44$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$40\!\cdots\!00$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$23\!\cdots\!03$$$$)/$$$$306110016$$ $$\nu^{8}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!65$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$46\!\cdots\!83$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$25\!\cdots\!83$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$42\!\cdots\!24$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$60\!\cdots\!92$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$33\!\cdots\!60$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$28\!\cdots\!13$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$19\!\cdots\!96$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$10\!\cdots\!91$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$24\!\cdots\!11$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$75\!\cdots\!42$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$93\!\cdots\!57$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$26\!\cdots\!36$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$29\!\cdots\!72$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$27\!\cdots\!44$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$15\!\cdots\!37$$$$)/$$$$3673320192$$ $$\nu^{9}$$ $$=$$ $$($$$$-$$$$51\!\cdots\!52$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$21\!\cdots\!88$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$14\!\cdots\!88$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$23\!\cdots\!10$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$14\!\cdots\!44$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$89\!\cdots\!68$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$44\!\cdots\!58$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$75\!\cdots\!10$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$12\!\cdots\!04$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$15\!\cdots\!54$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$13\!\cdots\!06$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$24\!\cdots\!07$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$32\!\cdots\!53$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$40\!\cdots\!22$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$18\!\cdots\!87$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$27\!\cdots\!87$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$45\!\cdots\!89$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$81\!\cdots\!32$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$21\!\cdots\!88$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$16\!\cdots\!15$$$$)/$$$$8264970432$$ $$\nu^{10}$$ $$=$$ $$($$$$54\!\cdots\!27$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$85\!\cdots\!33$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$44\!\cdots\!93$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$80\!\cdots\!08$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$12\!\cdots\!40$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$69\!\cdots\!44$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$59\!\cdots\!63$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$41\!\cdots\!52$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$16\!\cdots\!57$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$35\!\cdots\!01$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$16\!\cdots\!02$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$29\!\cdots\!23$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$44\!\cdots\!20$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$61\!\cdots\!60$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$56\!\cdots\!52$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$25\!\cdots\!99$$$$)/$$$$297538935552$$ $$\nu^{11}$$ $$=$$ $$($$$$97\!\cdots\!72$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$32\!\cdots\!48$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$29\!\cdots\!20$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$42\!\cdots\!66$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$27\!\cdots\!72$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$17\!\cdots\!64$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$94\!\cdots\!66$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$15\!\cdots\!34$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$33\!\cdots\!88$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$38\!\cdots\!86$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$25\!\cdots\!66$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$42\!\cdots\!29$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$63\!\cdots\!31$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$11\!\cdots\!62$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$50\!\cdots\!97$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$55\!\cdots\!01$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$89\!\cdots\!27$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$16\!\cdots\!08$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$36\!\cdots\!28$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$32\!\cdots\!41$$$$)/$$$$669462604992$$ $$\nu^{12}$$ $$=$$ $$($$$$-$$$$33\!\cdots\!39$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$51\!\cdots\!65$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$25\!\cdots\!85$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$50\!\cdots\!28$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$81\!\cdots\!80$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$45\!\cdots\!92$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$37\!\cdots\!19$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$26\!\cdots\!92$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$74\!\cdots\!45$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$13\!\cdots\!25$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$10\!\cdots\!74$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$64\!\cdots\!97$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$25\!\cdots\!64$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$42\!\cdots\!08$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$39\!\cdots\!28$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$14\!\cdots\!11$$$$)/$$$$8033551259904$$ $$\nu^{13}$$ $$=$$ $$($$$$-$$$$58\!\cdots\!80$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$14\!\cdots\!60$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$19\!\cdots\!08$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$23\!\cdots\!54$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$16\!\cdots\!92$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$11\!\cdots\!04$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$62\!\cdots\!82$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$98\!\cdots\!46$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$26\!\cdots\!16$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$30\!\cdots\!98$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$15\!\cdots\!90$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$24\!\cdots\!73$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$39\!\cdots\!95$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$88\!\cdots\!30$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$40\!\cdots\!73$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$36\!\cdots\!85$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$56\!\cdots\!23$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$10\!\cdots\!32$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$20\!\cdots\!40$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$20\!\cdots\!73$$$$)/$$$$18075490334784$$ $$\nu^{14}$$ $$=$$ $$($$$$22\!\cdots\!75$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$34\!\cdots\!69$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$16\!\cdots\!69$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$34\!\cdots\!92$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$56\!\cdots\!52$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$32\!\cdots\!16$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$24\!\cdots\!35$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$18\!\cdots\!08$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$32\!\cdots\!25$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$42\!\cdots\!13$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$74\!\cdots\!42$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$94\!\cdots\!05$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$15\!\cdots\!12$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$32\!\cdots\!24$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$29\!\cdots\!80$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$88\!\cdots\!99$$$$)/$$$$24100653779712$$ $$\nu^{15}$$ $$=$$ $$($$$$12\!\cdots\!12$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$18\!\cdots\!28$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$44\!\cdots\!00$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$49\!\cdots\!06$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$37\!\cdots\!12$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$27\!\cdots\!24$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$14\!\cdots\!06$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$22\!\cdots\!34$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$73\!\cdots\!28$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$80\!\cdots\!26$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$34\!\cdots\!06$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$51\!\cdots\!19$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$89\!\cdots\!25$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$24\!\cdots\!82$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$10\!\cdots\!51$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$85\!\cdots\!43$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$12\!\cdots\!25$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$23\!\cdots\!12$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$42\!\cdots\!48$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$46\!\cdots\!23$$$$)/$$$$18075490334784$$ $$\nu^{16}$$ $$=$$ $$($$$$-$$$$51\!\cdots\!81$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$76\!\cdots\!35$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$33\!\cdots\!15$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$79\!\cdots\!92$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$12\!\cdots\!16$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$75\!\cdots\!04$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$54\!\cdots\!17$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$42\!\cdots\!68$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$36\!\cdots\!07$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$16\!\cdots\!45$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$17\!\cdots\!22$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$31\!\cdots\!99$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$33\!\cdots\!80$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$78\!\cdots\!00$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$73\!\cdots\!80$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$18\!\cdots\!17$$$$)/$$$$24100653779712$$ $$\nu^{17}$$ $$=$$ $$($$$$-$$$$92\!\cdots\!40$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$47\!\cdots\!60$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$34\!\cdots\!52$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$33\!\cdots\!86$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$27\!\cdots\!48$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$21\!\cdots\!56$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$11\!\cdots\!38$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$17\!\cdots\!54$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$63\!\cdots\!24$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$68\!\cdots\!02$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$25\!\cdots\!30$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$36\!\cdots\!87$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$66\!\cdots\!89$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$21\!\cdots\!10$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$94\!\cdots\!91$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$66\!\cdots\!67$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$97\!\cdots\!45$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$17\!\cdots\!12$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$29\!\cdots\!40$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$35\!\cdots\!59$$$$)/$$$$6025163444928$$ $$\nu^{18}$$ $$=$$ $$($$$$37\!\cdots\!85$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$56\!\cdots\!11$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$23\!\cdots\!11$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$60\!\cdots\!68$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$95\!\cdots\!24$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$57\!\cdots\!20$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$38\!\cdots\!61$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$31\!\cdots\!72$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$22\!\cdots\!67$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$86\!\cdots\!13$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$12\!\cdots\!54$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$30\!\cdots\!31$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$23\!\cdots\!32$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$63\!\cdots\!64$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$59\!\cdots\!88$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$12\!\cdots\!29$$$$)/$$$$8033551259904$$ $$\nu^{19}$$ $$=$$ $$($$$$59\!\cdots\!84$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$24\!\cdots\!04$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$23\!\cdots\!16$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$20\!\cdots\!30$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$18\!\cdots\!28$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$15\!\cdots\!16$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$79\!\cdots\!66$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$11\!\cdots\!10$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$48\!\cdots\!08$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$51\!\cdots\!38$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$16\!\cdots\!02$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$23\!\cdots\!39$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$43\!\cdots\!61$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$15\!\cdots\!74$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$70\!\cdots\!59$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$46\!\cdots\!59$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$65\!\cdots\!53$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$11\!\cdots\!84$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$18\!\cdots\!96$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$23\!\cdots\!55$$$$)/$$$$18075490334784$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 − 4.67878e8i − 4.43955e8i − 3.58413e8i − 3.39530e8i − 3.00969e8i − 2.56585e8i − 2.39059e8i − 1.32287e8i − 1.26111e8i − 96378.2i 96378.2i 1.26111e8i 1.32287e8i 2.39059e8i 2.56585e8i 3.00969e8i 3.39530e8i 3.58413e8i 4.43955e8i 4.67878e8i
1.68436e10i −3.79959e15 + 4.05786e15i −2.09920e20 9.39164e22i 6.83490e25 + 6.39988e25i −5.61847e27 2.29297e30i −2.02937e30 3.08364e31i 1.58189e33
2.2 1.59824e10i 5.28212e15 1.73274e15i −1.81649e20 8.56960e22i −2.76933e25 8.44208e25i 1.24551e28 1.72390e30i 2.48984e31 1.83050e31i −1.36963e33
2.3 1.29029e10i −3.61968e15 4.21913e15i −9.26974e19 1.27679e23i −5.44390e25 + 4.67043e25i 1.78063e27 2.43999e29i −4.69903e30 + 3.05438e31i −1.64743e33
2.4 1.22231e10i 1.27232e15 5.41150e15i −7.56171e19 1.11898e23i −6.61453e25 1.55517e25i −1.34063e28 2.23699e28i −2.76656e31 1.37703e31i 1.36774e33
2.5 1.08349e10i 3.60729e15 + 4.22973e15i −4.36076e19 5.11119e22i 4.58286e25 3.90845e25i −1.55423e27 3.26990e29i −4.87811e30 + 3.05157e31i 5.53791e32
2.6 9.23704e9i −3.04579e15 + 4.65041e15i −1.15360e19 1.92130e23i 4.29561e25 + 2.81341e25i 2.12741e26 5.75015e29i −1.23495e31 2.83283e31i −1.77471e33
2.7 8.60612e9i −5.53379e15 5.29477e14i −2.78318e17 1.34902e23i −4.55674e24 + 4.76244e25i 8.41299e27 6.32624e29i 3.03425e31 + 5.86002e30i 1.16098e33
2.8 4.76234e9i 4.09693e15 3.75743e15i 5.11071e19 1.11626e23i −1.78942e25 1.95110e25i 1.00296e28 5.94788e29i 2.66653e30 3.07879e31i 5.31601e32
2.9 4.54000e9i 5.46530e15 1.01669e15i 5.31753e19 1.75307e23i −4.61577e24 2.48125e25i −8.19169e27 5.76409e29i 2.88358e31 1.11130e31i −7.95893e32
2.10 3.46962e6i −1.28962e15 5.40740e15i 7.37870e19 7.05043e22i −1.87616e22 + 4.47450e21i 2.25837e27 5.12025e26i −2.75769e31 + 1.39470e31i −2.44623e29
2.11 3.46962e6i −1.28962e15 + 5.40740e15i 7.37870e19 7.05043e22i −1.87616e22 4.47450e21i 2.25837e27 5.12025e26i −2.75769e31 1.39470e31i −2.44623e29
2.12 4.54000e9i 5.46530e15 + 1.01669e15i 5.31753e19 1.75307e23i −4.61577e24 + 2.48125e25i −8.19169e27 5.76409e29i 2.88358e31 + 1.11130e31i −7.95893e32
2.13 4.76234e9i 4.09693e15 + 3.75743e15i 5.11071e19 1.11626e23i −1.78942e25 + 1.95110e25i 1.00296e28 5.94788e29i 2.66653e30 + 3.07879e31i 5.31601e32
2.14 8.60612e9i −5.53379e15 + 5.29477e14i −2.78318e17 1.34902e23i −4.55674e24 4.76244e25i 8.41299e27 6.32624e29i 3.03425e31 5.86002e30i 1.16098e33
2.15 9.23704e9i −3.04579e15 4.65041e15i −1.15360e19 1.92130e23i 4.29561e25 2.81341e25i 2.12741e26 5.75015e29i −1.23495e31 + 2.83283e31i −1.77471e33
2.16 1.08349e10i 3.60729e15 4.22973e15i −4.36076e19 5.11119e22i 4.58286e25 + 3.90845e25i −1.55423e27 3.26990e29i −4.87811e30 3.05157e31i 5.53791e32
2.17 1.22231e10i 1.27232e15 + 5.41150e15i −7.56171e19 1.11898e23i −6.61453e25 + 1.55517e25i −1.34063e28 2.23699e28i −2.76656e31 + 1.37703e31i 1.36774e33
2.18 1.29029e10i −3.61968e15 + 4.21913e15i −9.26974e19 1.27679e23i −5.44390e25 4.67043e25i 1.78063e27 2.43999e29i −4.69903e30 3.05438e31i −1.64743e33
2.19 1.59824e10i 5.28212e15 + 1.73274e15i −1.81649e20 8.56960e22i −2.76933e25 + 8.44208e25i 1.24551e28 1.72390e30i 2.48984e31 + 1.83050e31i −1.36963e33
2.20 1.68436e10i −3.79959e15 4.05786e15i −2.09920e20 9.39164e22i 6.83490e25 6.39988e25i −5.61847e27 2.29297e30i −2.02937e30 + 3.08364e31i 1.58189e33
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{20} + \cdots$$ acting on $$S_{67}^{\mathrm{new}}(3, [\chi])$$.