# Properties

 Label 3.39.b.a Level $3$ Weight $39$ Character orbit 3.b Analytic conductor $27.439$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$39$$ Character orbit: $$[\chi]$$ $$=$$ 3.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.4390407101$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 17353504902 x^{10} + 111006258614054318328 x^{8} + 323765701965839203118204176384 x^{6} + 420150309279704216298413492838082805760 x^{4} + 190068212511425710374530430459662273636990976000 x^{2} + 27342285412416035125187079526375866471795145886924800000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{75}\cdot 3^{91}\cdot 5^{7}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -9561867 + 97 \beta_{1} + \beta_{2} ) q^{3} + ( -141606210704 - \beta_{2} + \beta_{3} ) q^{4} + ( -1705115 \beta_{1} - 340 \beta_{2} + \beta_{4} ) q^{5} + ( -40300933723344 - 5737864 \beta_{1} + 17 \beta_{2} + 707 \beta_{3} + 6 \beta_{4} - \beta_{7} ) q^{6} + ( 675656019711554 + 486451 \beta_{1} + 2066454 \beta_{2} + 346 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{7} + ( -106891803083 \beta_{1} - 45304936 \beta_{2} - 714 \beta_{3} + 1334 \beta_{4} - 9 \beta_{5} - 20 \beta_{7} + \beta_{8} ) q^{8} + ( -35359929288459495 - 284217479375 \beta_{1} - 1262720 \beta_{2} + 387217 \beta_{3} + 10480 \beta_{4} - 49 \beta_{5} - 10 \beta_{6} + 14 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q +\beta_{1} q^{2} +(-9561867 + 97 \beta_{1} + \beta_{2}) q^{3} +(-141606210704 - \beta_{2} + \beta_{3}) q^{4} +(-1705115 \beta_{1} - 340 \beta_{2} + \beta_{4}) q^{5} +(-40300933723344 - 5737864 \beta_{1} + 17 \beta_{2} + 707 \beta_{3} + 6 \beta_{4} - \beta_{7}) q^{6} +(675656019711554 + 486451 \beta_{1} + 2066454 \beta_{2} + 346 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7}) q^{7} +(-106891803083 \beta_{1} - 45304936 \beta_{2} - 714 \beta_{3} + 1334 \beta_{4} - 9 \beta_{5} - 20 \beta_{7} + \beta_{8}) q^{8} +(-35359929288459495 - 284217479375 \beta_{1} - 1262720 \beta_{2} + 387217 \beta_{3} + 10480 \beta_{4} - 49 \beta_{5} - 10 \beta_{6} + 14 \beta_{7} + 2 \beta_{8} + \beta_{9}) q^{9} +(710119790424444960 - 455618446 \beta_{1} - 1927629089 \beta_{2} - 8176408 \beta_{3} - 3179 \beta_{4} + 462 \beta_{5} + 194 \beta_{6} + 2527 \beta_{7} - \beta_{8} - 8 \beta_{9} + \beta_{11}) q^{10} +(27390754663596 \beta_{1} + 11652995750 \beta_{2} + 185972 \beta_{3} + 76601 \beta_{4} + 2341 \beta_{5} + 8 \beta_{6} + 4995 \beta_{7} - 293 \beta_{8} + 30 \beta_{9} + \beta_{10} + 3 \beta_{11}) q^{11} +(-238547044532722128 - 183484830441206 \beta_{1} - 136564546404 \beta_{2} - 150439761 \beta_{3} + 3388702 \beta_{4} - 11135 \beta_{5} - 14442 \beta_{6} + 3832 \beta_{7} + 1722 \beta_{8} - 72 \beta_{9} + 15 \beta_{10} + 9 \beta_{11}) q^{12} +(89091564444452625146 + 55263541281 \beta_{1} + 235496387523 \beta_{2} - 697666076 \beta_{3} - 134562 \beta_{4} - 27096 \beta_{5} + 84600 \beta_{6} + 668568 \beta_{7} + 329 \beta_{8} + 472 \beta_{9} + 135 \beta_{10} - 59 \beta_{11}) q^{13} +(600532489034223 \beta_{1} - 249327308600 \beta_{2} - 3440546 \beta_{3} - 20034253 \beta_{4} + 47672 \beta_{5} - 4076 \beta_{6} - 5541125 \beta_{7} - 1266 \beta_{8} - 1920 \beta_{9} + 827 \beta_{10} - 192 \beta_{11}) q^{14} +($$$$52\!\cdots\!40$$$$+ 6245149471880296 \beta_{1} - 186133268171 \beta_{2} - 17104287632 \beta_{3} + 347123922 \beta_{4} + 365178 \beta_{5} - 947259 \beta_{6} + 1557463 \beta_{7} - 54909 \beta_{8} + 2538 \beta_{9} + 3825 \beta_{10} - 621 \beta_{11}) q^{15} +($$$$55\!\cdots\!88$$$$- 2494188410618 \beta_{1} - 10479902478018 \beta_{2} - 117396179496 \beta_{3} - 55711960 \beta_{4} + 787524 \beta_{5} + 479380 \beta_{6} + 94176272 \beta_{7} + 25078 \beta_{8} - 12784 \beta_{9} + 13338 \beta_{10} + 1598 \beta_{11}) q^{16} +(-56025599754603189 \beta_{1} - 42748812501027 \beta_{2} - 672767318 \beta_{3} - 2889669704 \beta_{4} - 5179330 \beta_{5} - 114200 \beta_{6} - 218023564 \beta_{7} + 608119 \beta_{8} + 57750 \beta_{9} + 34325 \beta_{10} + 5775 \beta_{11}) q^{17} +($$$$11\!\cdots\!60$$$$- 128353304280714909 \beta_{1} - 35888093280609 \beta_{2} - 866153505228 \beta_{3} + 28821558693 \beta_{4} + 1244670 \beta_{5} + 34011114 \beta_{6} + 11978979 \beta_{7} + 802851 \beta_{8} - 58344 \beta_{9} + 52380 \beta_{10} + 20493 \beta_{11}) q^{18} +(-$$$$38\!\cdots\!30$$$$- 28220154698801 \beta_{1} - 122468212881275 \beta_{2} + 2568729226482 \beta_{3} - 4231165 \beta_{4} - 4345587 \beta_{5} - 77021570 \beta_{6} - 155809681 \beta_{7} - 24564 \beta_{8} + 206112 \beta_{9} - 25164 \beta_{10} - 25764 \beta_{11}) q^{19} +(2197892044416010190 \beta_{1} + 1259633825834640 \beta_{2} + 18372761060 \beta_{3} - 251726070196 \beta_{4} + 187040770 \beta_{5} + 1488800 \beta_{6} + 3200020480 \beta_{7} - 16660930 \beta_{8} - 1075200 \beta_{9} - 479720 \beta_{10} - 107520 \beta_{11}) q^{20} +($$$$27\!\cdots\!46$$$$- 747998542258694032 \beta_{1} + 717046895828107 \beta_{2} - 15312896577090 \beta_{3} + 248149340371 \beta_{4} - 22506098 \beta_{5} - 344144652 \beta_{6} - 974793740 \beta_{7} + 1763409 \beta_{8} + 981378 \beta_{9} - 1739181 \beta_{10} - 429399 \beta_{11}) q^{21} +(-$$$$11\!\cdots\!80$$$$+ 440436341457781 \beta_{1} + 1763408432235986 \beta_{2} + 107928692108020 \beta_{3} + 13560597245 \beta_{4} + 664168632 \beta_{5} + 1377885028 \beta_{6} - 25842855229 \beta_{7} - 7865138 \beta_{8} - 2101984 \beta_{9} - 3801195 \beta_{10} + 262748 \beta_{11}) q^{22} +(-26787086683905899794 \beta_{1} - 87257900318298 \beta_{2} - 828702692 \beta_{3} + 665830890518 \beta_{4} - 442538926 \beta_{5} + 21788800 \beta_{6} + 25425965682 \beta_{7} + 234600272 \beta_{8} + 13668000 \beta_{9} - 4080400 \beta_{10} + 1366800 \beta_{11}) q^{23} +($$$$65\!\cdots\!32$$$$+ 33791945754583867135 \beta_{1} - 638290082267978 \beta_{2} - 492514336252346 \beta_{3} - 3233275975530 \beta_{4} + 10096449639 \beta_{5} - 460338156 \beta_{6} + 27638759212 \beta_{7} - 252024597 \beta_{8} - 12820464 \beta_{9} + 5030298 \beta_{10} + 6392574 \beta_{11}) q^{24} +(-$$$$10\!\cdots\!35$$$$- 28443364351272709 \beta_{1} - 121377002786677411 \beta_{2} + 528314122722748 \beta_{3} - 243688213946 \beta_{4} + 17122514628 \beta_{5} - 9680661364 \beta_{6} + 273595097968 \beta_{7} + 75496811 \beta_{8} + 12266248 \beta_{9} + 36981765 \beta_{10} - 1533281 \beta_{11}) q^{25} +($$$$25\!\cdots\!82$$$$\beta_{1} + 288297764264519200 \beta_{2} + 4788185111448 \beta_{3} + 22771577555084 \beta_{4} + 71216921424 \beta_{5} - 425984304 \beta_{6} - 525459889140 \beta_{7} - 2140307832 \beta_{8} - 122008320 \beta_{9} + 94295244 \beta_{10} - 12200832 \beta_{11}) q^{26} +($$$$29\!\cdots\!17$$$$-$$$$24\!\cdots\!23$$$$\beta_{1} - 26344517184065691 \beta_{2} - 2010091010267556 \beta_{3} - 23150651113323 \beta_{4} + 70887934017 \beta_{5} + 41182835742 \beta_{6} - 132762058947 \beta_{7} + 4244825997 \beta_{8} + 134463330 \beta_{9} + 140225175 \beta_{10} - 71623035 \beta_{11}) q^{27} +(-$$$$64\!\cdots\!64$$$$- 336556120571587042 \beta_{1} - 1430394628658130668 \beta_{2} + 456785202946266 \beta_{3} - 1646137011896 \beta_{4} + 300597846612 \beta_{5} - 2484509212 \beta_{6} + 548972805328 \beta_{7} + 71382094 \beta_{8} - 6194608 \beta_{9} + 36078210 \beta_{10} + 774326 \beta_{11}) q^{28} +(-$$$$10\!\cdots\!27$$$$\beta_{1} + 2905747846092856550 \beta_{2} + 45665656678372 \beta_{3} - 65627849239509 \beta_{4} + 543397999436 \beta_{5} + 1900316944 \beta_{6} + 3087523112040 \beta_{7} + 12749585702 \beta_{8} + 731245020 \beta_{9} - 401954734 \beta_{10} + 73124502 \beta_{11}) q^{29} +(-$$$$26\!\cdots\!20$$$$+$$$$46\!\cdots\!87$$$$\beta_{1} + 558586540417140898 \beta_{2} + 18860294575218816 \beta_{3} + 174017843828123 \beta_{4} + 1902563908256 \beta_{5} - 402613028148 \beta_{6} + 3716965156241 \beta_{7} - 38850812238 \beta_{8} - 1151856864 \beta_{9} - 1323199005 \beta_{10} + 623197548 \beta_{11}) q^{30} +($$$$52\!\cdots\!82$$$$- 2524872460226435945 \beta_{1} - 10707541626430795140 \beta_{2} - 19945971849264902 \beta_{3} - 4627687572952 \beta_{4} + 2812946199462 \beta_{5} + 561725838487 \beta_{6} - 12540081785041 \beta_{7} - 4504315596 \beta_{8} - 636667680 \beta_{9} - 2212366068 \beta_{10} + 79583460 \beta_{11}) q^{31} +($$$$43\!\cdots\!28$$$$\beta_{1} + 27716359898457440000 \beta_{2} + 434473199020480 \beta_{3} - 1089116262972352 \beta_{4} + 5367848715488 \beta_{5} + 7055526400 \beta_{6} + 17841577607936 \beta_{7} - 33094777568 \beta_{8} - 2145792000 \beta_{9} - 1978460800 \beta_{10} - 214579200 \beta_{11}) q^{32} +(-$$$$16\!\cdots\!20$$$$-$$$$59\!\cdots\!04$$$$\beta_{1} + 82469908697798411 \beta_{2} + 136345866128782373 \beta_{3} + 873830119658700 \beta_{4} + 5493557745051 \beta_{5} + 1653083816922 \beta_{6} - 6985489227154 \beta_{7} + 225514914471 \beta_{8} + 8100337257 \beta_{9} + 2053426635 \beta_{10} - 4273162479 \beta_{11}) q^{33} +($$$$23\!\cdots\!92$$$$- 17656281543064870852 \beta_{1} - 74786314788783110904 \beta_{2} - 230637631317283728 \beta_{3} - 111435459222836 \beta_{4} + 12931834300032 \beta_{5} - 4289186198704 \beta_{6} + 85171093040500 \beta_{7} + 21481371224 \beta_{8} + 6709291264 \beta_{9} + 10321354908 \beta_{10} - 838661408 \beta_{11}) q^{34} +(-$$$$26\!\cdots\!35$$$$\beta_{1} + 87410197297432307715 \beta_{2} + 1434849438702790 \beta_{3} + 4537769335210254 \beta_{4} + 20797400700980 \beta_{5} - 96758372600 \beta_{6} - 125784393587630 \beta_{7} - 224004072645 \beta_{8} - 9185974050 \beta_{9} + 23270995745 \beta_{10} - 918597405 \beta_{11}) q^{35} +($$$$43\!\cdots\!12$$$$+$$$$24\!\cdots\!78$$$$\beta_{1} - 7436031932689446869 \beta_{2} - 434823822556177595 \beta_{3} - 7610473251121052 \beta_{4} + 34929411081014 \beta_{5} + 744774337688 \beta_{6} + 97193913857120 \beta_{7} - 843299988922 \beta_{8} - 46428255392 \beta_{9} + 27246173796 \beta_{10} + 23077495188 \beta_{11}) q^{36} +(-$$$$87\!\cdots\!46$$$$- 15158815315448853689 \beta_{1} - 64984693657887025427 \beta_{2} + 578717592374995740 \beta_{3} - 125984631923926 \beta_{4} + 16718934734928 \beta_{5} + 11330062646320 \beta_{6} + 121238547255800 \beta_{7} + 26724583887 \beta_{8} - 36050751384 \beta_{9} + 15615463905 \beta_{10} + 4506343923 \beta_{11}) q^{37} +(-$$$$10\!\cdots\!63$$$$\beta_{1} -$$$$18\!\cdots\!76$$$$\beta_{2} - 2876924533857614 \beta_{3} + 18660478413653325 \beta_{4} - 42141675757408 \beta_{5} + 267392848300 \beta_{6} + 272195666340597 \beta_{7} + 3225129786170 \beta_{8} + 166179552000 \beta_{9} - 50230256875 \beta_{10} + 16617955200 \beta_{11}) q^{38} +($$$$31\!\cdots\!14$$$$+$$$$95\!\cdots\!05$$$$\beta_{1} + 62623026919039607065 \beta_{2} - 1830692252264024610 \beta_{3} - 11694291806969726 \beta_{4} - 115962178410020 \beta_{5} - 40185901161870 \beta_{6} - 323912344487636 \beta_{7} + 1819867478475 \beta_{8} + 211321005066 \beta_{9} - 135706363671 \beta_{10} - 96033816261 \beta_{11}) q^{39} +(-$$$$72\!\cdots\!20$$$$+$$$$26\!\cdots\!52$$$$\beta_{1} +$$$$11\!\cdots\!48$$$$\beta_{2} + 6756095367820761616 \beta_{3} + 2125702709474928 \beta_{4} - 192549350989224 \beta_{5} + 40368305011512 \beta_{6} - 2158813038055584 \beta_{7} - 553444275228 \beta_{8} + 75592984416 \beta_{9} - 281446699140 \beta_{10} - 9449123052 \beta_{11}) q^{40} +(-$$$$19\!\cdots\!46$$$$\beta_{1} -$$$$79\!\cdots\!00$$$$\beta_{2} - 13613493799806032 \beta_{3} - 141011002948988106 \beta_{4} - 198567824538496 \beta_{5} + 493056035392 \beta_{6} + 1322572308335280 \beta_{7} - 19780150505992 \beta_{8} - 1187163904080 \beta_{9} - 241980399256 \beta_{10} - 118716390408 \beta_{11}) q^{41} +($$$$31\!\cdots\!40$$$$+$$$$64\!\cdots\!16$$$$\beta_{1} -$$$$19\!\cdots\!43$$$$\beta_{2} - 2148665495400002360 \beta_{3} + 164745771961641903 \beta_{4} - 583820068344654 \beta_{5} + 153050465846286 \beta_{6} - 167607787451075 \beta_{7} - 1341244240263 \beta_{8} - 711458034552 \beta_{9} - 136829229660 \beta_{10} + 286711315119 \beta_{11}) q^{42} +($$$$83\!\cdots\!66$$$$+$$$$86\!\cdots\!87$$$$\beta_{1} +$$$$36\!\cdots\!53$$$$\beta_{2} - 12495565869512929502 \beta_{3} + 1908490221361135 \beta_{4} - 1069547529267111 \beta_{5} - 461895699745198 \beta_{6} + 3867020126885095 \beta_{7} + 1419058886188 \beta_{8} + 463218481184 \beta_{9} + 680578288020 \beta_{10} - 57902310148 \beta_{11}) q^{43} +(-$$$$29\!\cdots\!46$$$$\beta_{1} -$$$$10\!\cdots\!00$$$$\beta_{2} - 173910237892624236 \beta_{3} + 106283723373376732 \beta_{4} - 2110216876523238 \beta_{5} - 2724094726368 \beta_{6} - 9619173592042240 \beta_{7} + 72536523618854 \beta_{8} + 5445352857600 \beta_{9} + 1225558967352 \beta_{10} + 544535285760 \beta_{11}) q^{44} +(-$$$$51\!\cdots\!60$$$$+$$$$32\!\cdots\!71$$$$\beta_{1} -$$$$48\!\cdots\!86$$$$\beta_{2} + 30033770122900592898 \beta_{3} + 19034514037355343 \beta_{4} - 1775596873836642 \beta_{5} + 196046033543076 \beta_{6} - 2306025181110372 \beta_{7} + 4271281132386 \beta_{8} + 1352999538618 \beta_{9} + 2787878238930 \beta_{10} - 457147899306 \beta_{11}) q^{45} +($$$$11\!\cdots\!72$$$$+$$$$18\!\cdots\!14$$$$\beta_{1} +$$$$80\!\cdots\!40$$$$\beta_{2} - 81376745937518586760 \beta_{3} + 2709638422055318 \beta_{4} - 1139454577885440 \beta_{5} + 1764374780357032 \beta_{6} + 8706136014327818 \beta_{7} + 2324383068460 \beta_{8} - 5310291906688 \beta_{9} + 1494084778398 \beta_{10} + 663786488336 \beta_{11}) q^{46} +(-$$$$64\!\cdots\!34$$$$\beta_{1} +$$$$42\!\cdots\!54$$$$\beta_{2} + 74758531848771276 \beta_{3} + 892410525149630932 \beta_{4} + 1157216128763904 \beta_{5} - 12886450772400 \beta_{6} - 10537138412171604 \beta_{7} - 147126022209522 \beta_{8} - 16050909388500 \beta_{9} + 1616521754250 \beta_{10} - 1605090938850 \beta_{11}) q^{47} +(-$$$$14\!\cdots\!72$$$$+$$$$13\!\cdots\!86$$$$\beta_{1} +$$$$15\!\cdots\!30$$$$\beta_{2} + 99531776001858523128 \beta_{3} - 2124875254989240312 \beta_{4} + 2718801568311348 \beta_{5} - 3975252245642844 \beta_{6} + 4746264541199184 \beta_{7} - 98840760709554 \beta_{8} + 1903238532432 \beta_{9} - 6794078328990 \beta_{10} - 718191011754 \beta_{11}) q^{48} +(-$$$$61\!\cdots\!21$$$$-$$$$61\!\cdots\!23$$$$\beta_{1} -$$$$26\!\cdots\!53$$$$\beta_{2} + 90093475731030478084 \beta_{3} - 466009434656150 \beta_{4} + 5213840999004204 \beta_{5} - 3068805972181180 \beta_{6} - 48270638648648528 \beta_{7} - 11170513450187 \beta_{8} + 26595095593976 \beta_{9} - 7247450199717 \beta_{10} - 3324386949247 \beta_{11}) q^{49} +(-$$$$23\!\cdots\!75$$$$\beta_{1} +$$$$76\!\cdots\!00$$$$\beta_{2} + 1200328930483382600 \beta_{3} - 588114373631702140 \beta_{4} + 12882556994765200 \beta_{5} + 102440129942000 \beta_{6} + 165992297466879300 \beta_{7} + 39215756774200 \beta_{8} + 19380317088000 \beta_{9} - 23672000776700 \beta_{10} + 1938031708800 \beta_{11}) q^{50} +($$$$59\!\cdots\!56$$$$+$$$$24\!\cdots\!49$$$$\beta_{1} -$$$$68\!\cdots\!29$$$$\beta_{2} -$$$$34\!\cdots\!18$$$$\beta_{3} + 1831987056703869444 \beta_{4} + 19426258943982210 \beta_{5} + 16464632256433476 \beta_{6} + 8923669398503160 \beta_{7} + 709210238748345 \beta_{8} - 27412496825022 \beta_{9} - 5889017078613 \beta_{10} + 7519233795201 \beta_{11}) q^{51} +(-$$$$82\!\cdots\!36$$$$-$$$$39\!\cdots\!16$$$$\beta_{1} -$$$$16\!\cdots\!62$$$$\beta_{2} +$$$$71\!\cdots\!82$$$$\beta_{3} - 124629025234012000 \beta_{4} + 39054288517870608 \beta_{5} + 1511031414630736 \beta_{6} - 82570578907201984 \beta_{7} - 51104182856616 \beta_{8} - 74856417098688 \beta_{9} - 20873565359640 \beta_{10} + 9357052137336 \beta_{11}) q^{52} +(-$$$$46\!\cdots\!59$$$$\beta_{1} +$$$$16\!\cdots\!68$$$$\beta_{2} + 2703413397641146592 \beta_{3} - 4160417948812300011 \beta_{4} + 40129043841417088 \beta_{5} - 187182042428800 \beta_{6} - 296904500480807328 \beta_{7} + 567735159735856 \beta_{8} + 78520460268000 \beta_{9} + 54647556634000 \beta_{10} + 7852046026800 \beta_{11}) q^{53} +($$$$10\!\cdots\!76$$$$+$$$$77\!\cdots\!43$$$$\beta_{1} +$$$$96\!\cdots\!83$$$$\beta_{2} -$$$$91\!\cdots\!81$$$$\beta_{3} + 14575059318925273077 \beta_{4} + 65817985452264792 \beta_{5} - 30830997264616692 \beta_{6} + 82871160255572976 \beta_{7} - 2677065627425526 \beta_{8} + 114930327265632 \beta_{9} + 24773713192215 \beta_{10} - 24969189777228 \beta_{11}) q^{54} +(-$$$$12\!\cdots\!20$$$$-$$$$72\!\cdots\!68$$$$\beta_{1} -$$$$30\!\cdots\!22$$$$\beta_{2} -$$$$76\!\cdots\!04$$$$\beta_{3} - 876370688186817442 \beta_{4} + 39596480344456206 \beta_{5} - 12107524707375578 \beta_{6} + 1215585412129465736 \beta_{7} + 354782478934372 \beta_{8} + 51785712569696 \beta_{9} + 174154632431580 \beta_{10} - 6473214071212 \beta_{11}) q^{55} +(-$$$$14\!\cdots\!26$$$$\beta_{1} +$$$$12\!\cdots\!00$$$$\beta_{2} + 1944033416166409452 \beta_{3} - 22940044212684888724 \beta_{4} + 26810074507111086 \beta_{5} - 218881997004288 \beta_{6} - 79406889571609000 \beta_{7} - 832578525812158 \beta_{8} - 523816538664960 \beta_{9} + 2338845384576 \beta_{10} - 52381653866496 \beta_{11}) q^{56} +(-$$$$16\!\cdots\!26$$$$-$$$$69\!\cdots\!37$$$$\beta_{1} -$$$$36\!\cdots\!66$$$$\beta_{2} +$$$$13\!\cdots\!47$$$$\beta_{3} - 10720770471791443806 \beta_{4} - 69339373608465879 \beta_{5} + 10139080081292514 \beta_{6} - 15753481488856734 \beta_{7} + 4634021768157720 \beta_{8} - 241699131335889 \beta_{9} + 200407318391430 \beta_{10} + 37636363499058 \beta_{11}) q^{57} +($$$$45\!\cdots\!60$$$$+$$$$30\!\cdots\!90$$$$\beta_{1} +$$$$12\!\cdots\!29$$$$\beta_{2} + 63172056881629030744 \beta_{3} + 2535677486567438607 \beta_{4} - 179607368891188422 \beta_{5} + 114498399724472214 \beta_{6} - 2802201655865445459 \beta_{7} - 623442040703403 \beta_{8} + 565252625651496 \beta_{9} - 347049309454920 \beta_{10} - 70656578206437 \beta_{11}) q^{58} +($$$$19\!\cdots\!41$$$$\beta_{1} -$$$$14\!\cdots\!75$$$$\beta_{2} - 22031066367683383086 \beta_{3} +$$$$13\!\cdots\!37$$$$\beta_{4} - 285930444497395293 \beta_{5} + 499626168471648 \beta_{6} - 90695937773248245 \beta_{7} - 2235889322236476 \beta_{8} + 1405973646824040 \beta_{9} + 15690822564492 \beta_{10} + 140597364682404 \beta_{11}) q^{59} +(-$$$$17\!\cdots\!40$$$$-$$$$54\!\cdots\!66$$$$\beta_{1} +$$$$15\!\cdots\!56$$$$\beta_{2} +$$$$13\!\cdots\!32$$$$\beta_{3} - 91227594899698143732 \beta_{4} - 539875008744244878 \beta_{5} + 23113649893586184 \beta_{6} - 787016368293871328 \beta_{7} + 6445790383381794 \beta_{8} - 74964336984288 \beta_{9} - 1048730428146420 \beta_{10} + 37600148077596 \beta_{11}) q^{60} +($$$$16\!\cdots\!02$$$$+$$$$35\!\cdots\!79$$$$\beta_{1} +$$$$15\!\cdots\!01$$$$\beta_{2} -$$$$13\!\cdots\!36$$$$\beta_{3} + 1019935112357887458 \beta_{4} - 506645254346652000 \beta_{5} - 357521963541645408 \beta_{6} + 1279136699199772728 \beta_{7} - 166140380919285 \beta_{8} - 2822005804299768 \beta_{9} + 93305172309093 \beta_{10} + 352750725537471 \beta_{11}) q^{61} +($$$$99\!\cdots\!47$$$$\beta_{1} -$$$$34\!\cdots\!00$$$$\beta_{2} - 57556895200063011530 \beta_{3} -$$$$15\!\cdots\!13$$$$\beta_{4} - 877452289986409048 \beta_{5} + 5120793845827300 \beta_{6} + 7678687812883043719 \beta_{7} + 3538052233391478 \beta_{8} - 1117447527504000 \beta_{9} - 1391943214207225 \beta_{10} - 111744752750400 \beta_{11}) q^{62} +(-$$$$57\!\cdots\!74$$$$-$$$$94\!\cdots\!03$$$$\beta_{1} +$$$$30\!\cdots\!22$$$$\beta_{2} -$$$$49\!\cdots\!86$$$$\beta_{3} -$$$$10\!\cdots\!58$$$$\beta_{4} - 482285613665874152 \beta_{5} + 222334394104982407 \beta_{6} - 325467892732554839 \beta_{7} - 58921230724949744 \beta_{8} + 2439961354397624 \beta_{9} + 1622694819546720 \beta_{10} - 342088777592928 \beta_{11}) q^{63} +(-$$$$27\!\cdots\!12$$$$+$$$$14\!\cdots\!12$$$$\beta_{1} +$$$$60\!\cdots\!56$$$$\beta_{2} +$$$$19\!\cdots\!84$$$$\beta_{3} + 9158277165005273344 \beta_{4} - 1088968962053109120 \beta_{5} + 219979573345533056 \beta_{6} - 7150080416835155456 \beta_{7} - 503371017401920 \beta_{8} + 5866869003536896 \beta_{9} - 618364821422016 \beta_{10} - 733358625442112 \beta_{11}) q^{64} +($$$$24\!\cdots\!60$$$$\beta_{1} -$$$$54\!\cdots\!90$$$$\beta_{2} - 81756758019520175140 \beta_{3} +$$$$14\!\cdots\!46$$$$\beta_{4} - 526948722811241180 \beta_{5} - 21498728881056400 \beta_{6} - 32418437617294808920 \beta_{7} + 41150227727358570 \beta_{8} - 5846058930752700 \beta_{9} + 4790076327188830 \beta_{10} - 584605893075270 \beta_{11}) q^{65} +($$$$24\!\cdots\!60$$$$-$$$$49\!\cdots\!06$$$$\beta_{1} -$$$$99\!\cdots\!59$$$$\beta_{2} -$$$$48\!\cdots\!60$$$$\beta_{3} +$$$$13\!\cdots\!35$$$$\beta_{4} - 197782199295584534 \beta_{5} - 755691724975692066 \beta_{6} + 244831326568005277 \beta_{7} + 134824150269941817 \beta_{8} - 8430656186013048 \beta_{9} - 208343883664056 \beta_{10} + 780155291329695 \beta_{11}) q^{66} +(-$$$$10\!\cdots\!46$$$$-$$$$22\!\cdots\!49$$$$\beta_{1} -$$$$94\!\cdots\!53$$$$\beta_{2} +$$$$18\!\cdots\!34$$$$\beta_{3} - 31705948987609575179 \beta_{4} + 1899292757953481391 \beta_{5} + 1582737385835595704 \beta_{6} + 45465232985726136631 \beta_{7} + 13578292804390184 \beta_{8} + 1124999330149312 \beta_{9} + 6718833944060760 \beta_{10} - 140624916268664 \beta_{11}) q^{67} +($$$$63\!\cdots\!28$$$$\beta_{1} +$$$$30\!\cdots\!96$$$$\beta_{2} +$$$$46\!\cdots\!04$$$$\beta_{3} -$$$$16\!\cdots\!84$$$$\beta_{4} + 5452009054847675544 \beta_{5} + 27060809732707200 \beta_{6} + 39689938566465141760 \beta_{7} - 206882408252316056 \beta_{8} + 24619512766464000 \beta_{9} - 4303251156530400 \beta_{10} + 2461951276646400 \beta_{11}) q^{68} +($$$$11\!\cdots\!36$$$$-$$$$25\!\cdots\!86$$$$\beta_{1} +$$$$76\!\cdots\!52$$$$\beta_{2} -$$$$10\!\cdots\!98$$$$\beta_{3} -$$$$14\!\cdots\!16$$$$\beta_{4} + 6005268658627722366 \beta_{5} - 974937900775454532 \beta_{6} + 7985070294640904332 \beta_{7} - 24325846511114268 \beta_{8} + 10838488398257250 \beta_{9} + 3098690042367456 \beta_{10} - 441952937029152 \beta_{11}) q^{69} +($$$$10\!\cdots\!40$$$$-$$$$12\!\cdots\!94$$$$\beta_{1} -$$$$54\!\cdots\!76$$$$\beta_{2} +$$$$93\!\cdots\!68$$$$\beta_{3} - 15763879093163102286 \beta_{4} + 11235689431928230248 \beta_{5} - 4346784729603047424 \beta_{6} - 71677185794394518562 \beta_{7} - 34953216382218624 \beta_{8} - 43958596612580832 \beta_{9} - 14729195902823010 \beta_{10} + 5494824576572604 \beta_{11}) q^{70} +(-$$$$39\!\cdots\!08$$$$\beta_{1} +$$$$36\!\cdots\!00$$$$\beta_{2} +$$$$60\!\cdots\!84$$$$\beta_{3} +$$$$45\!\cdots\!42$$$$\beta_{4} + 7509892586223450802 \beta_{5} + 2656369230596816 \beta_{6} + 23988027142116280190 \beta_{7} + 364915602799650654 \beta_{8} - 36945261899698740 \beta_{9} - 4358618497619078 \beta_{10} - 3694526189969874 \beta_{11}) q^{71} +(-$$$$70\!\cdots\!20$$$$+$$$$11\!\cdots\!01$$$$\beta_{1} +$$$$61\!\cdots\!24$$$$\beta_{2} +$$$$34\!\cdots\!90$$$$\beta_{3} -$$$$15\!\cdots\!58$$$$\beta_{4} + 15179344269259437255 \beta_{5} + 9188354804625235272 \beta_{6} + 3041023411307784108 \beta_{7} - 566271965983289739 \beta_{8} + 23367021289685664 \beta_{9} - 21592375516945980 \beta_{10} - 2082999760641108 \beta_{11}) q^{72} +($$$$75\!\cdots\!06$$$$-$$$$76\!\cdots\!24$$$$\beta_{1} -$$$$32\!\cdots\!20$$$$\beta_{2} -$$$$36\!\cdots\!64$$$$\beta_{3} + 20939277311717259504 \beta_{4} + 2681964765545325768 \beta_{5} + 2411080195838277912 \beta_{6} - 52613010242253875184 \beta_{7} + 10087878439308060 \beta_{8} + 123372592306757280 \beta_{9} - 2666847799518300 \beta_{10} - 15421574038344660 \beta_{11}) q^{73} +(-$$$$22\!\cdots\!62$$$$\beta_{1} +$$$$20\!\cdots\!00$$$$\beta_{2} +$$$$31\!\cdots\!76$$$$\beta_{3} -$$$$31\!\cdots\!32$$$$\beta_{4} + 3556377166064866928 \beta_{5} + 12887592775144624 \beta_{6} + 31750645977040560660 \beta_{7} + 294310271633419256 \beta_{8} - 25040345045295360 \beta_{9} - 5725932698315692 \beta_{10} - 2504034504529536 \beta_{11}) q^{74} +(-$$$$16\!\cdots\!55$$$$+$$$$22\!\cdots\!18$$$$\beta_{1} -$$$$11\!\cdots\!78$$$$\beta_{2} +$$$$16\!\cdots\!54$$$$\beta_{3} -$$$$75\!\cdots\!18$$$$\beta_{4} - 8426695866525378056 \beta_{5} - 16039576997140990722 \beta_{6} + 2504164414031285164 \beta_{7} + 1160806382715533853 \beta_{8} - 140263792625727846 \beta_{9} + 18295864382834295 \beta_{10} + 5786446917852837 \beta_{11}) q^{75} +($$$$31\!\cdots\!04$$$$+$$$$66\!\cdots\!06$$$$\beta_{1} +$$$$29\!\cdots\!48$$$$\beta_{2} -$$$$13\!\cdots\!82$$$$\beta_{3} -$$$$11\!\cdots\!72$$$$\beta_{4} - 10405544156186550516 \beta_{5} + 2408785413924912892 \beta_{6} +$$$$28\!\cdots\!60$$$$\beta_{7} + 69588135609007938 \beta_{8} - 91669491523779792 \beta_{9} + 40523411024740206 \beta_{10} + 11458686440472474 \beta_{11}) q^{76} +(-$$$$38\!\cdots\!50$$$$\beta_{1} -$$$$74\!\cdots\!72$$$$\beta_{2} -$$$$11\!\cdots\!68$$$$\beta_{3} +$$$$62\!\cdots\!74$$$$\beta_{4} - 11658887362955365872 \beta_{5} - 73073343527140800 \beta_{6} -$$$$19\!\cdots\!28$$$$\beta_{7} - 2961643936073331304 \beta_{8} + 217040902725798000 \beta_{9} + 39972426154365000 \beta_{10} + 21704090272579800 \beta_{11}) q^{77} +(-$$$$39\!\cdots\!40$$$$+$$$$75\!\cdots\!36$$$$\beta_{1} -$$$$25\!\cdots\!66$$$$\beta_{2} +$$$$35\!\cdots\!94$$$$\beta_{3} +$$$$36\!\cdots\!64$$$$\beta_{4} - 61078282624573004424 \beta_{5} - 5063965185948868296 \beta_{6} - 15072815295054677342 \beta_{7} - 176734958034565836 \beta_{8} + 260958836585360160 \beta_{9} + 86749142556798000 \beta_{10} - 4786657700354820 \beta_{11}) q^{78} +($$$$28\!\cdots\!10$$$$+$$$$74\!\cdots\!35$$$$\beta_{1} +$$$$31\!\cdots\!84$$$$\beta_{2} +$$$$93\!\cdots\!10$$$$\beta_{3} +$$$$29\!\cdots\!04$$$$\beta_{4} - 61660497701867049834 \beta_{5} + 17157072733501385751 \beta_{6} + 3066987996005151327 \beta_{7} - 57627220182269428 \beta_{8} - 392544096295021280 \beta_{9} - 4279604072695884 \beta_{10} + 49068012036877660 \beta_{11}) q^{79} +(-$$$$17\!\cdots\!40$$$$\beta_{1} -$$$$76\!\cdots\!40$$$$\beta_{2} -$$$$12\!\cdots\!40$$$$\beta_{3} -$$$$35\!\cdots\!64$$$$\beta_{4} -$$$$14\!\cdots\!80$$$$\beta_{5} - 435801327012275200 \beta_{6} -$$$$74\!\cdots\!20$$$$\beta_{7} + 6205492116181312320 \beta_{8} - 364254272767795200 \beta_{9} + 72524904476289280 \beta_{10} - 36425427276779520 \beta_{11}) q^{80} +(-$$$$32\!\cdots\!99$$$$+$$$$12\!\cdots\!91$$$$\beta_{1} +$$$$25\!\cdots\!83$$$$\beta_{2} -$$$$27\!\cdots\!26$$$$\beta_{3} -$$$$22\!\cdots\!98$$$$\beta_{4} -$$$$11\!\cdots\!70$$$$\beta_{5} + 42657455380906010304 \beta_{6} -$$$$21\!\cdots\!44$$$$\beta_{7} - 1657859039803007955 \beta_{8} + 68071671254003526 \beta_{9} - 172956392440634241 \beta_{10} - 3368231980945875 \beta_{11}) q^{81} +($$$$80\!\cdots\!80$$$$+$$$$17\!\cdots\!24$$$$\beta_{1} +$$$$75\!\cdots\!54$$$$\beta_{2} +$$$$23\!\cdots\!40$$$$\beta_{3} +$$$$10\!\cdots\!10$$$$\beta_{4} -$$$$17\!\cdots\!52$$$$\beta_{5} - 48402833926886053428 \beta_{6} -$$$$51\!\cdots\!26$$$$\beta_{7} + 164338500670149978 \beta_{8} + 1337727884598538704 \beta_{9} - 1438742452333680 \beta_{10} - 167215985574817338 \beta_{11}) q^{82} +(-$$$$87\!\cdots\!94$$$$\beta_{1} -$$$$19\!\cdots\!96$$$$\beta_{2} -$$$$31\!\cdots\!24$$$$\beta_{3} +$$$$30\!\cdots\!67$$$$\beta_{4} - 84072274568348558981 \beta_{5} + 1661438543014077800 \beta_{6} +$$$$25\!\cdots\!01$$$$\beta_{7} - 2000413324208028737 \beta_{8} - 3333721986152250 \beta_{9} - 415693007952134675 \beta_{10} - 333372198615225 \beta_{11}) q^{83} +(-$$$$19\!\cdots\!96$$$$+$$$$62\!\cdots\!14$$$$\beta_{1} -$$$$96\!\cdots\!54$$$$\beta_{2} -$$$$41\!\cdots\!94$$$$\beta_{3} -$$$$19\!\cdots\!48$$$$\beta_{4} + 62957472971940386214 \beta_{5} - 4347123589805681376 \beta_{6} - 14660291841428469504 \beta_{7} - 1760789278045235142 \beta_{8} - 1481256978939562752 \beta_{9} + 2541238745994216 \beta_{10} + 3681080010103392 \beta_{11}) q^{84} +($$$$13\!\cdots\!80$$$$-$$$$11\!\cdots\!48$$$$\beta_{1} -$$$$50\!\cdots\!72$$$$\beta_{2} +$$$$10\!\cdots\!36$$$$\beta_{3} +$$$$94\!\cdots\!08$$$$\beta_{4} +$$$$12\!\cdots\!96$$$$\beta_{5} - 86342973182505462848 \beta_{6} -$$$$28\!\cdots\!44$$$$\beta_{7} - 1183333381558445468 \beta_{8} - 1287478595333896864 \beta_{9} - 511199278570854180 \beta_{10} + 160934824416737108 \beta_{11}) q^{85} +($$$$38\!\cdots\!77$$$$\beta_{1} +$$$$18\!\cdots\!00$$$$\beta_{2} +$$$$30\!\cdots\!86$$$$\beta_{3} +$$$$55\!\cdots\!53$$$$\beta_{4} +$$$$41\!\cdots\!48$$$$\beta_{5} - 1023584600263944724 \beta_{6} -$$$$14\!\cdots\!75$$$$\beta_{7} - 18690744099459295494 \beta_{8} + 922302013645989120 \beta_{9} + 348126351430585093 \beta_{10} + 92230201364598912 \beta_{11}) q^{86} +(-$$$$38\!\cdots\!60$$$$-$$$$41\!\cdots\!06$$$$\beta_{1} +$$$$96\!\cdots\!61$$$$\beta_{2} -$$$$24\!\cdots\!64$$$$\beta_{3} -$$$$11\!\cdots\!74$$$$\beta_{4} +$$$$16\!\cdots\!34$$$$\beta_{5} - 39893969886115808529 \beta_{6} +$$$$62\!\cdots\!57$$$$\beta_{7} + 7751791536844796811 \beta_{8} + 3163606543386373410 \beta_{9} - 221083588311755175 \beta_{10} + 8302337076094155 \beta_{11}) q^{87} +($$$$92\!\cdots\!60$$$$-$$$$78\!\cdots\!60$$$$\beta_{1} -$$$$33\!\cdots\!36$$$$\beta_{2} -$$$$30\!\cdots\!96$$$$\beta_{3} -$$$$73\!\cdots\!88$$$$\beta_{4} +$$$$77\!\cdots\!48$$$$\beta_{5} +$$$$45\!\cdots\!24$$$$\beta_{6} +$$$$80\!\cdots\!56$$$$\beta_{7} + 1817439218666837652 \beta_{8} - 2263203642083909664 \beta_{9} + 1050169836963663180 \beta_{10} + 282900455260488708 \beta_{11}) q^{88} +($$$$40\!\cdots\!29$$$$\beta_{1} +$$$$10\!\cdots\!25$$$$\beta_{2} +$$$$16\!\cdots\!66$$$$\beta_{3} +$$$$81\!\cdots\!58$$$$\beta_{4} +$$$$25\!\cdots\!58$$$$\beta_{5} - 1538371133881750488 \beta_{6} -$$$$21\!\cdots\!80$$$$\beta_{7} + 45751752574214014831 \beta_{8} - 1107769609956002490 \beta_{9} + 273815822474837373 \beta_{10} - 110776960995600249 \beta_{11}) q^{89} +(-$$$$13\!\cdots\!60$$$$-$$$$12\!\cdots\!54$$$$\beta_{1} -$$$$23\!\cdots\!21$$$$\beta_{2} +$$$$26\!\cdots\!68$$$$\beta_{3} +$$$$19\!\cdots\!29$$$$\beta_{4} +$$$$94\!\cdots\!98$$$$\beta_{5} -$$$$35\!\cdots\!74$$$$\beta_{6} +$$$$72\!\cdots\!43$$$$\beta_{7} + 16568600347234737531 \beta_{8} - 684637618646821032 \beta_{9} + 1552204917009964080 \beta_{10} + 47889780034672629 \beta_{11}) q^{90} +($$$$10\!\cdots\!52$$$$-$$$$46\!\cdots\!18$$$$\beta_{1} -$$$$19\!\cdots\!76$$$$\beta_{2} +$$$$69\!\cdots\!88$$$$\beta_{3} -$$$$19\!\cdots\!24$$$$\beta_{4} +$$$$28\!\cdots\!88$$$$\beta_{5} -$$$$30\!\cdots\!86$$$$\beta_{6} +$$$$34\!\cdots\!50$$$$\beta_{7} + 1589586621437926916 \beta_{8} + 7978462216265620576 \beta_{9} + 296139422202362172 \beta_{10} - 997307777033202572 \beta_{11}) q^{91} +($$$$23\!\cdots\!88$$$$\beta_{1} +$$$$70\!\cdots\!52$$$$\beta_{2} +$$$$11\!\cdots\!68$$$$\beta_{3} -$$$$27\!\cdots\!76$$$$\beta_{4} +$$$$15\!\cdots\!40$$$$\beta_{5} - 3321015155834075200 \beta_{6} -$$$$32\!\cdots\!16$$$$\beta_{7} - 42727588872565877804 \beta_{8} - 108498958356480000 \beta_{9} + 819403893122870800 \beta_{10} - 10849895835648000 \beta_{11}) q^{92} +(-$$$$14\!\cdots\!94$$$$-$$$$10\!\cdots\!78$$$$\beta_{1} +$$$$54\!\cdots\!29$$$$\beta_{2} -$$$$18\!\cdots\!36$$$$\beta_{3} +$$$$28\!\cdots\!33$$$$\beta_{4} +$$$$12\!\cdots\!00$$$$\beta_{5} +$$$$99\!\cdots\!48$$$$\beta_{6} -$$$$39\!\cdots\!24$$$$\beta_{7} - 86359228748735269353 \beta_{8} - 10801252449283229856 \beta_{9} - 883297079755976055 \beta_{10} - 245879019004889493 \beta_{11}) q^{93} +($$$$26\!\cdots\!72$$$$-$$$$12\!\cdots\!44$$$$\beta_{1} -$$$$54\!\cdots\!92$$$$\beta_{2} -$$$$33\!\cdots\!16$$$$\beta_{3} -$$$$14\!\cdots\!36$$$$\beta_{4} +$$$$87\!\cdots\!48$$$$\beta_{5} -$$$$10\!\cdots\!24$$$$\beta_{6} -$$$$71\!\cdots\!12$$$$\beta_{7} - 4017477871813732624 \beta_{8} - 6420246176507959232 \beta_{9} - 1607473549875118860 \beta_{10} + 802530772063494904 \beta_{11}) q^{94} +($$$$23\!\cdots\!30$$$$\beta_{1} -$$$$39\!\cdots\!70$$$$\beta_{2} -$$$$69\!\cdots\!40$$$$\beta_{3} -$$$$82\!\cdots\!02$$$$\beta_{4} -$$$$11\!\cdots\!30$$$$\beta_{5} + 9280077863292564800 \beta_{6} +$$$$14\!\cdots\!30$$$$\beta_{7} - 28614154357394751080 \beta_{8} - 1346001090646621200 \beta_{9} - 2454619574887803320 \beta_{10} - 134600109064662120 \beta_{11}) q^{95} +(-$$$$37\!\cdots\!52$$$$-$$$$28\!\cdots\!20$$$$\beta_{1} +$$$$60\!\cdots\!24$$$$\beta_{2} +$$$$45\!\cdots\!36$$$$\beta_{3} -$$$$33\!\cdots\!40$$$$\beta_{4} -$$$$24\!\cdots\!88$$$$\beta_{5} + 39749769779997057408 \beta_{6} +$$$$33\!\cdots\!00$$$$\beta_{7} + 81708084566305350624 \beta_{8} + 22961930417607292416 \beta_{9} - 4299813119615350464 \beta_{10} + 254912493521003328 \beta_{11}) q^{96} +($$$$20\!\cdots\!94$$$$+$$$$47\!\cdots\!17$$$$\beta_{1} +$$$$20\!\cdots\!55$$$$\beta_{2} +$$$$14\!\cdots\!36$$$$\beta_{3} +$$$$26\!\cdots\!02$$$$\beta_{4} -$$$$36\!\cdots\!28$$$$\beta_{5} +$$$$12\!\cdots\!64$$$$\beta_{6} -$$$$14\!\cdots\!20$$$$\beta_{7} - 4640269810231590983 \beta_{8} - 7943436718327595944 \beta_{9} - 1823670110220320745 \beta_{10} + 992929589790949493 \beta_{11}) q^{97} +(-$$$$27\!\cdots\!49$$$$\beta_{1} -$$$$22\!\cdots\!56$$$$\beta_{2} -$$$$34\!\cdots\!64$$$$\beta_{3} +$$$$17\!\cdots\!32$$$$\beta_{4} -$$$$48\!\cdots\!56$$$$\beta_{5} + 21774413176593232400 \beta_{6} +$$$$23\!\cdots\!56$$$$\beta_{7} +$$$$20\!\cdots\!08$$$$\beta_{8} + 8283881081096544000 \beta_{9} - 4615215186038653700 \beta_{10} + 828388108109654400 \beta_{11}) q^{98} +(-$$$$28\!\cdots\!20$$$$-$$$$50\!\cdots\!29$$$$\beta_{1} -$$$$16\!\cdots\!27$$$$\beta_{2} -$$$$21\!\cdots\!38$$$$\beta_{3} +$$$$21\!\cdots\!11$$$$\beta_{4} -$$$$83\!\cdots\!51$$$$\beta_{5} -$$$$28\!\cdots\!86$$$$\beta_{6} -$$$$18\!\cdots\!09$$$$\beta_{7} +$$$$15\!\cdots\!98$$$$\beta_{8} - 6535349611762124004 \beta_{9} + 4049332131129180498 \beta_{10} + 539071471116633654 \beta_{11}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 114742404q^{3} - 1699274528448q^{4} - 483611204680128q^{6} + 8107872236538648q^{7} - 424319151461513940q^{9} + O(q^{10})$$ $$12q - 114742404q^{3} - 1699274528448q^{4} - 483611204680128q^{6} + 8107872236538648q^{7} - 424319151461513940q^{9} + 8521437485093339520q^{10} - 2862564534392665536q^{12} +$$$$10\!\cdots\!52$$$$q^{13} +$$$$63\!\cdots\!80$$$$q^{15} +$$$$67\!\cdots\!56$$$$q^{16} +$$$$14\!\cdots\!20$$$$q^{18} -$$$$46\!\cdots\!60$$$$q^{19} +$$$$33\!\cdots\!52$$$$q^{21} -$$$$13\!\cdots\!60$$$$q^{22} +$$$$78\!\cdots\!84$$$$q^{24} -$$$$12\!\cdots\!20$$$$q^{25} +$$$$35\!\cdots\!04$$$$q^{27} -$$$$77\!\cdots\!68$$$$q^{28} -$$$$31\!\cdots\!40$$$$q^{30} +$$$$62\!\cdots\!84$$$$q^{31} -$$$$20\!\cdots\!40$$$$q^{33} +$$$$27\!\cdots\!04$$$$q^{34} +$$$$52\!\cdots\!44$$$$q^{36} -$$$$10\!\cdots\!52$$$$q^{37} +$$$$37\!\cdots\!68$$$$q^{39} -$$$$86\!\cdots\!40$$$$q^{40} +$$$$37\!\cdots\!80$$$$q^{42} +$$$$10\!\cdots\!92$$$$q^{43} -$$$$61\!\cdots\!20$$$$q^{45} +$$$$13\!\cdots\!64$$$$q^{46} -$$$$16\!\cdots\!64$$$$q^{48} -$$$$74\!\cdots\!52$$$$q^{49} +$$$$71\!\cdots\!72$$$$q^{51} -$$$$99\!\cdots\!32$$$$q^{52} +$$$$12\!\cdots\!12$$$$q^{54} -$$$$14\!\cdots\!40$$$$q^{55} -$$$$19\!\cdots\!12$$$$q^{57} +$$$$54\!\cdots\!20$$$$q^{58} -$$$$21\!\cdots\!80$$$$q^{60} +$$$$19\!\cdots\!24$$$$q^{61} -$$$$68\!\cdots\!88$$$$q^{63} -$$$$33\!\cdots\!44$$$$q^{64} +$$$$29\!\cdots\!20$$$$q^{66} -$$$$12\!\cdots\!52$$$$q^{67} +$$$$14\!\cdots\!32$$$$q^{69} +$$$$13\!\cdots\!80$$$$q^{70} -$$$$84\!\cdots\!40$$$$q^{72} +$$$$90\!\cdots\!72$$$$q^{73} -$$$$19\!\cdots\!60$$$$q^{75} +$$$$37\!\cdots\!48$$$$q^{76} -$$$$47\!\cdots\!80$$$$q^{78} +$$$$33\!\cdots\!20$$$$q^{79} -$$$$38\!\cdots\!88$$$$q^{81} +$$$$97\!\cdots\!60$$$$q^{82} -$$$$23\!\cdots\!52$$$$q^{84} +$$$$16\!\cdots\!60$$$$q^{85} -$$$$46\!\cdots\!20$$$$q^{87} +$$$$11\!\cdots\!20$$$$q^{88} -$$$$16\!\cdots\!20$$$$q^{90} +$$$$12\!\cdots\!24$$$$q^{91} -$$$$17\!\cdots\!28$$$$q^{93} +$$$$32\!\cdots\!64$$$$q^{94} -$$$$45\!\cdots\!24$$$$q^{96} +$$$$24\!\cdots\!28$$$$q^{97} -$$$$34\!\cdots\!40$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 17353504902 x^{10} + 111006258614054318328 x^{8} + 323765701965839203118204176384 x^{6} + 420150309279704216298413492838082805760 x^{4} + 190068212511425710374530430459662273636990976000 x^{2} + 27342285412416035125187079526375866471795145886924800000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$12 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$64\!\cdots\!25$$$$\nu^{11} +$$$$19\!\cdots\!92$$$$\nu^{10} -$$$$89\!\cdots\!50$$$$\nu^{9} +$$$$30\!\cdots\!44$$$$\nu^{8} -$$$$37\!\cdots\!00$$$$\nu^{7} +$$$$16\!\cdots\!56$$$$\nu^{6} -$$$$44\!\cdots\!00$$$$\nu^{5} +$$$$37\!\cdots\!88$$$$\nu^{4} +$$$$25\!\cdots\!00$$$$\nu^{3} +$$$$32\!\cdots\!20$$$$\nu^{2} +$$$$21\!\cdots\!00$$$$\nu +$$$$74\!\cdots\!00$$$$)/$$$$21\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$64\!\cdots\!25$$$$\nu^{11} +$$$$19\!\cdots\!92$$$$\nu^{10} -$$$$89\!\cdots\!50$$$$\nu^{9} +$$$$30\!\cdots\!44$$$$\nu^{8} -$$$$37\!\cdots\!00$$$$\nu^{7} +$$$$16\!\cdots\!56$$$$\nu^{6} -$$$$44\!\cdots\!00$$$$\nu^{5} +$$$$37\!\cdots\!88$$$$\nu^{4} +$$$$25\!\cdots\!00$$$$\nu^{3} +$$$$63\!\cdots\!20$$$$\nu^{2} +$$$$21\!\cdots\!00$$$$\nu +$$$$96\!\cdots\!00$$$$)/$$$$21\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!73$$$$\nu^{11} +$$$$66\!\cdots\!28$$$$\nu^{10} -$$$$19\!\cdots\!66$$$$\nu^{9} +$$$$10\!\cdots\!96$$$$\nu^{8} -$$$$10\!\cdots\!24$$$$\nu^{7} +$$$$55\!\cdots\!04$$$$\nu^{6} -$$$$24\!\cdots\!12$$$$\nu^{5} +$$$$12\!\cdots\!92$$$$\nu^{4} -$$$$20\!\cdots\!00$$$$\nu^{3} +$$$$11\!\cdots\!80$$$$\nu^{2} -$$$$46\!\cdots\!00$$$$\nu +$$$$25\!\cdots\!00$$$$)/$$$$21\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$54\!\cdots\!91$$$$\nu^{11} -$$$$93\!\cdots\!48$$$$\nu^{10} -$$$$32\!\cdots\!62$$$$\nu^{9} -$$$$14\!\cdots\!36$$$$\nu^{8} +$$$$30\!\cdots\!12$$$$\nu^{7} -$$$$78\!\cdots\!64$$$$\nu^{6} +$$$$26\!\cdots\!76$$$$\nu^{5} -$$$$17\!\cdots\!72$$$$\nu^{4} +$$$$53\!\cdots\!40$$$$\nu^{3} -$$$$15\!\cdots\!80$$$$\nu^{2} +$$$$19\!\cdots\!00$$$$\nu -$$$$36\!\cdots\!00$$$$)/$$$$21\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$38\!\cdots\!31$$$$\nu^{11} +$$$$25\!\cdots\!96$$$$\nu^{10} -$$$$64\!\cdots\!42$$$$\nu^{9} +$$$$29\!\cdots\!52$$$$\nu^{8} -$$$$40\!\cdots\!08$$$$\nu^{7} +$$$$70\!\cdots\!68$$$$\nu^{6} -$$$$11\!\cdots\!84$$$$\nu^{5} -$$$$16\!\cdots\!76$$$$\nu^{4} -$$$$13\!\cdots\!60$$$$\nu^{3} -$$$$59\!\cdots\!40$$$$\nu^{2} -$$$$38\!\cdots\!00$$$$\nu -$$$$18\!\cdots\!00$$$$)/$$$$53\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!49$$$$\nu^{11} -$$$$17\!\cdots\!32$$$$\nu^{10} -$$$$30\!\cdots\!18$$$$\nu^{9} -$$$$25\!\cdots\!24$$$$\nu^{8} -$$$$16\!\cdots\!32$$$$\nu^{7} -$$$$12\!\cdots\!76$$$$\nu^{6} -$$$$36\!\cdots\!36$$$$\nu^{5} -$$$$21\!\cdots\!48$$$$\nu^{4} -$$$$32\!\cdots\!40$$$$\nu^{3} -$$$$10\!\cdots\!20$$$$\nu^{2} -$$$$73\!\cdots\!00$$$$\nu -$$$$86\!\cdots\!00$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$75\!\cdots\!71$$$$\nu^{11} -$$$$14\!\cdots\!92$$$$\nu^{10} +$$$$12\!\cdots\!22$$$$\nu^{9} -$$$$19\!\cdots\!44$$$$\nu^{8} +$$$$76\!\cdots\!28$$$$\nu^{7} -$$$$81\!\cdots\!56$$$$\nu^{6} +$$$$20\!\cdots\!44$$$$\nu^{5} -$$$$81\!\cdots\!88$$$$\nu^{4} +$$$$23\!\cdots\!60$$$$\nu^{3} +$$$$85\!\cdots\!80$$$$\nu^{2} +$$$$83\!\cdots\!00$$$$\nu +$$$$45\!\cdots\!00$$$$)/$$$$21\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$25\!\cdots\!11$$$$\nu^{11} -$$$$10\!\cdots\!92$$$$\nu^{10} +$$$$47\!\cdots\!82$$$$\nu^{9} -$$$$15\!\cdots\!44$$$$\nu^{8} +$$$$31\!\cdots\!88$$$$\nu^{7} -$$$$77\!\cdots\!56$$$$\nu^{6} +$$$$93\!\cdots\!84$$$$\nu^{5} -$$$$16\!\cdots\!88$$$$\nu^{4} +$$$$11\!\cdots\!60$$$$\nu^{3} -$$$$11\!\cdots\!20$$$$\nu^{2} +$$$$29\!\cdots\!00$$$$\nu -$$$$24\!\cdots\!00$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$10\!\cdots\!23$$$$\nu^{11} -$$$$31\!\cdots\!28$$$$\nu^{10} +$$$$15\!\cdots\!86$$$$\nu^{9} -$$$$41\!\cdots\!56$$$$\nu^{8} +$$$$84\!\cdots\!64$$$$\nu^{7} -$$$$15\!\cdots\!84$$$$\nu^{6} +$$$$19\!\cdots\!72$$$$\nu^{5} -$$$$90\!\cdots\!52$$$$\nu^{4} +$$$$16\!\cdots\!80$$$$\nu^{3} +$$$$33\!\cdots\!20$$$$\nu^{2} +$$$$38\!\cdots\!00$$$$\nu +$$$$14\!\cdots\!00$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$24\!\cdots\!09$$$$\nu^{11} +$$$$10\!\cdots\!92$$$$\nu^{10} +$$$$43\!\cdots\!78$$$$\nu^{9} +$$$$15\!\cdots\!04$$$$\nu^{8} +$$$$28\!\cdots\!32$$$$\nu^{7} +$$$$77\!\cdots\!36$$$$\nu^{6} +$$$$81\!\cdots\!16$$$$\nu^{5} +$$$$16\!\cdots\!48$$$$\nu^{4} +$$$$96\!\cdots\!40$$$$\nu^{3} +$$$$11\!\cdots\!20$$$$\nu^{2} +$$$$24\!\cdots\!00$$$$\nu +$$$$24\!\cdots\!00$$$$)/$$$$10\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/12$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} - 416484117648$$$$)/144$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{8} - 20 \beta_{7} - 9 \beta_{5} + 1334 \beta_{4} - 714 \beta_{3} - 45304936 \beta_{2} - 656647616971 \beta_{1}$$$$)/1728$$ $$\nu^{4}$$ $$=$$ $$($$$$799 \beta_{11} + 6669 \beta_{10} - 6392 \beta_{9} + 12539 \beta_{8} + 47088136 \beta_{7} + 239690 \beta_{6} + 393762 \beta_{5} - 27855980 \beta_{4} - 471014950164 \beta_{3} - 4827634378593 \beta_{2} - 1247094205309 \beta_{1} + 136739430988927591014144$$$$)/10368$$ $$\nu^{5}$$ $$=$$ $$($$$$-6705600 \beta_{11} - 61826900 \beta_{10} - 67056000 \beta_{9} - 35393950167 \beta_{8} + 1244744067608 \beta_{7} + 220485200 \beta_{6} + 476982917671 \beta_{5} - 79870774200798 \beta_{4} + 38110140664142 \beta_{3} + 2422801994565779448 \beta_{2} + 15615161793150985633113 \beta_{1}$$$$)/7776$$ $$\nu^{6}$$ $$=$$ $$($$$$-15258505739191 \beta_{11} - 98697773101653 \beta_{10} + 122068045913528 \beta_{9} - 182137040464115 \beta_{8} - 711380016003017288 \beta_{7} - 2285808759419482 \beta_{6} - 11309029718718450 \beta_{5} + 446501103894502732 \beta_{4} + 4483183904191038657684 \beta_{3} + 102758841196879549578601 \beta_{2} + 25240662805316933219141 \beta_{1} - 1083871560824682019823285236088064$$$$)/15552$$ $$\nu^{7}$$ $$=$$ $$($$$$99182555428440000 \beta_{11} + 1295065687396294900 \beta_{10} + 991825554284400000 \beta_{9} + 370338136631142908741 \beta_{8} - 17757837781737448706576 \beta_{7} - 4783532527871419600 \beta_{6} - 5678078049686331088325 \beta_{5} + 1096167248320355444491314 \beta_{4} - 457513123140565387678042 \beta_{3} - 29154280914062865880015909288 \beta_{2} - 135012583980859288782195363234163 \beta_{1}$$$$)/11664$$ $$\nu^{8}$$ $$=$$ $$($$$$69148968752239466894431 \beta_{11} + 374894423822411950483149 \beta_{10} - 553191750017915735155448 \beta_{9} + 680639878892584434071867 \beta_{8} + 2749462141706947022269585160 \beta_{7} + 4157713660772753478694858 \beta_{6} + 60501768950149151114291106 \beta_{5} - 1806837164947397906187781228 \beta_{4} - 14407333215307329975466032759348 \beta_{3} - 486246632118159252660015069138433 \beta_{2} - 117834985733986370272472760793661 \beta_{1} + 3123763265555227088202800914376907762102528$$$$)/7776$$ $$\nu^{9}$$ $$=$$ $$($$$$-371589767032584553967870400 \beta_{11} - 6164654658599298190175218900 \beta_{10} - 3715897670325845539678704000 \beta_{9} - 1249291362867536527678100670425 \beta_{8} + 72196003546279773341550204010688 \beta_{7} + 23172259566266854544829394000 \beta_{6} + 20151151573164475635090447385913 \beta_{5} - 4339552662811867207569098126290890 \beta_{4} + 1635499767933901056655909541062114 \beta_{3} + 104428226042747080567456360838090341576 \beta_{2} + 411383115161410030859426714103269736329071 \beta_{1}$$$$)/5832$$ $$\nu^{10}$$ $$=$$ $$($$$$-273041063278624275941834669396875 \beta_{11} - 1313014395647663854412599843497489 \beta_{10} + 2184328506228994207534677355175000 \beta_{9} - 2352987728016703432883365017598103 \beta_{8} - 9760741401779474047109913658691068328 \beta_{7} - 1478140447780421818629323710322194 \beta_{6} - 260251364933100926765875323286341834 \beta_{5} + 6634906906959240143382121353263887324 \beta_{4} + 46796177626378833306322987770502871970756 \beta_{3} + 1965079398013998004116379965297211633849813 \beta_{2} + 473519845695187263487624596803892524782145 \beta_{1} - 9518016447981516609234775917271798602015576485350656$$$$)/3888$$ $$\nu^{11}$$ $$=$$ $$($$$$1279958726284170746771894545002820800 \beta_{11} + 25055191514492707769075876664609561700 \beta_{10} + 12799587262841707467718945450028208000 \beta_{9} + 4164213936055426159129739292336300830741 \beta_{8} - 270597421775670663354467628769238163610208 \beta_{7} - 95100931152834148089215928478426963600 \beta_{6} - 68674282138302235967361310669630354408757 \beta_{5} + 15979945773202597880720600545664681609947074 \beta_{4} - 5605180436972689518626551019810478093365338 \beta_{3} - 358445096981947526973357701700936725761055459752 \beta_{2} - 1296852824569760987584387021355100020786032868160307 \beta_{1}$$$$)/2916$$

## 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
 − 81459.4i − 65373.8i − 54008.5i − 53338.0i − 18842.7i − 18089.7i 18089.7i 18842.7i 53338.0i 54008.5i 65373.8i 81459.4i
977513.i −4.43499e8 1.07432e9i −6.80653e11 1.94075e13i −1.05016e15 + 4.33526e14i −1.85443e15 3.96650e17i −9.57469e17 + 9.52919e17i 1.89711e19
2.2 784486.i 1.15997e9 + 7.30136e7i −3.40540e11 3.07374e13i 5.72781e13 9.09976e14i 1.07708e16 5.15107e16i 1.34019e18 + 1.69386e17i −2.41131e19
2.3 648102.i −9.89523e8 + 6.09669e8i −1.45158e11 1.81665e13i 3.95128e14 + 6.41312e14i −1.77541e16 8.40715e16i 6.07458e17 1.20656e18i −1.17737e19
2.4 640056.i 2.88152e8 + 1.12598e9i −1.34793e11 3.15820e13i 7.20687e14 1.84433e14i 9.50531e15 8.96619e16i −1.18479e18 + 6.48904e17i 2.02142e19
2.5 226112.i −8.54601e8 7.87723e8i 2.23751e11 6.89065e12i −1.78114e14 + 1.93236e14i 1.19822e16 1.12746e17i 1.09835e17 + 1.34638e18i −1.55806e18
2.6 217076.i 7.82134e8 8.59720e8i 2.27756e11 1.16100e13i −1.86625e14 1.69783e14i −8.59590e15 1.09110e17i −1.27385e17 1.34483e18i 2.52026e18
2.7 217076.i 7.82134e8 + 8.59720e8i 2.27756e11 1.16100e13i −1.86625e14 + 1.69783e14i −8.59590e15 1.09110e17i −1.27385e17 + 1.34483e18i 2.52026e18
2.8 226112.i −8.54601e8 + 7.87723e8i 2.23751e11 6.89065e12i −1.78114e14 1.93236e14i 1.19822e16 1.12746e17i 1.09835e17 1.34638e18i −1.55806e18
2.9 640056.i 2.88152e8 1.12598e9i −1.34793e11 3.15820e13i 7.20687e14 + 1.84433e14i 9.50531e15 8.96619e16i −1.18479e18 6.48904e17i 2.02142e19
2.10 648102.i −9.89523e8 6.09669e8i −1.45158e11 1.81665e13i 3.95128e14 6.41312e14i −1.77541e16 8.40715e16i 6.07458e17 + 1.20656e18i −1.17737e19
2.11 784486.i 1.15997e9 7.30136e7i −3.40540e11 3.07374e13i 5.72781e13 + 9.09976e14i 1.07708e16 5.15107e16i 1.34019e18 1.69386e17i −2.41131e19
2.12 977513.i −4.43499e8 + 1.07432e9i −6.80653e11 1.94075e13i −1.05016e15 4.33526e14i −1.85443e15 3.96650e17i −9.57469e17 9.52919e17i 1.89711e19
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.39.b.a 12
3.b odd 2 1 inner 3.39.b.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.39.b.a 12 1.a even 1 1 trivial
3.39.b.a 12 3.b odd 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{39}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 799630177440 T^{2} +$$$$41\!\cdots\!64$$$$T^{4} -$$$$16\!\cdots\!80$$$$T^{6} +$$$$53\!\cdots\!20$$$$T^{8} -$$$$15\!\cdots\!80$$$$T^{10} +$$$$43\!\cdots\!60$$$$T^{12} -$$$$11\!\cdots\!80$$$$T^{14} +$$$$30\!\cdots\!20$$$$T^{16} -$$$$72\!\cdots\!80$$$$T^{18} +$$$$13\!\cdots\!24$$$$T^{20} -$$$$19\!\cdots\!40$$$$T^{22} +$$$$18\!\cdots\!36$$$$T^{24}$$
$3$ $$1 + 114742404 T + 218742485368606578 T^{2} -$$$$11\!\cdots\!44$$$$T^{3} +$$$$84\!\cdots\!79$$$$T^{4} -$$$$20\!\cdots\!68$$$$T^{5} +$$$$15\!\cdots\!24$$$$T^{6} -$$$$27\!\cdots\!52$$$$T^{7} +$$$$15\!\cdots\!59$$$$T^{8} -$$$$28\!\cdots\!36$$$$T^{9} +$$$$72\!\cdots\!98$$$$T^{10} +$$$$51\!\cdots\!96$$$$T^{11} +$$$$60\!\cdots\!61$$$$T^{12}$$
$5$ $$1 -$$$$15\!\cdots\!40$$$$T^{2} +$$$$13\!\cdots\!50$$$$T^{4} -$$$$88\!\cdots\!00$$$$T^{6} +$$$$47\!\cdots\!75$$$$T^{8} -$$$$21\!\cdots\!00$$$$T^{10} +$$$$85\!\cdots\!00$$$$T^{12} -$$$$28\!\cdots\!00$$$$T^{14} +$$$$83\!\cdots\!75$$$$T^{16} -$$$$20\!\cdots\!00$$$$T^{18} +$$$$41\!\cdots\!50$$$$T^{20} -$$$$62\!\cdots\!00$$$$T^{22} +$$$$53\!\cdots\!25$$$$T^{24}$$
$7$ $$( 1 - 4053936118269324 T +$$$$41\!\cdots\!98$$$$T^{2} -$$$$33\!\cdots\!76$$$$T^{3} +$$$$90\!\cdots\!99$$$$T^{4} +$$$$61\!\cdots\!48$$$$T^{5} +$$$$13\!\cdots\!04$$$$T^{6} +$$$$79\!\cdots\!52$$$$T^{7} +$$$$15\!\cdots\!99$$$$T^{8} -$$$$73\!\cdots\!24$$$$T^{9} +$$$$11\!\cdots\!98$$$$T^{10} -$$$$15\!\cdots\!76$$$$T^{11} +$$$$48\!\cdots\!01$$$$T^{12} )^{2}$$
$11$ $$1 -$$$$13\!\cdots\!32$$$$T^{2} +$$$$71\!\cdots\!26$$$$T^{4} -$$$$13\!\cdots\!20$$$$T^{6} -$$$$30\!\cdots\!05$$$$T^{8} +$$$$30\!\cdots\!08$$$$T^{10} -$$$$12\!\cdots\!36$$$$T^{12} +$$$$41\!\cdots\!88$$$$T^{14} -$$$$60\!\cdots\!05$$$$T^{16} -$$$$38\!\cdots\!20$$$$T^{18} +$$$$27\!\cdots\!66$$$$T^{20} -$$$$72\!\cdots\!32$$$$T^{22} +$$$$75\!\cdots\!61$$$$T^{24}$$
$13$ $$( 1 -$$$$53\!\cdots\!76$$$$T +$$$$59\!\cdots\!58$$$$T^{2} -$$$$45\!\cdots\!04$$$$T^{3} +$$$$21\!\cdots\!59$$$$T^{4} -$$$$16\!\cdots\!68$$$$T^{5} +$$$$55\!\cdots\!44$$$$T^{6} -$$$$35\!\cdots\!72$$$$T^{7} +$$$$99\!\cdots\!19$$$$T^{8} -$$$$44\!\cdots\!56$$$$T^{9} +$$$$12\!\cdots\!98$$$$T^{10} -$$$$23\!\cdots\!24$$$$T^{11} +$$$$95\!\cdots\!21$$$$T^{12} )^{2}$$
$17$ $$1 -$$$$25\!\cdots\!60$$$$T^{2} +$$$$44\!\cdots\!34$$$$T^{4} -$$$$53\!\cdots\!20$$$$T^{6} +$$$$51\!\cdots\!35$$$$T^{8} -$$$$39\!\cdots\!20$$$$T^{10} +$$$$24\!\cdots\!00$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{14} +$$$$55\!\cdots\!35$$$$T^{16} -$$$$18\!\cdots\!20$$$$T^{18} +$$$$51\!\cdots\!14$$$$T^{20} -$$$$95\!\cdots\!60$$$$T^{22} +$$$$12\!\cdots\!81$$$$T^{24}$$
$19$ $$( 1 +$$$$23\!\cdots\!80$$$$T +$$$$21\!\cdots\!94$$$$T^{2} +$$$$38\!\cdots\!60$$$$T^{3} +$$$$19\!\cdots\!55$$$$T^{4} +$$$$27\!\cdots\!60$$$$T^{5} +$$$$97\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!60$$$$T^{7} +$$$$29\!\cdots\!55$$$$T^{8} +$$$$22\!\cdots\!60$$$$T^{9} +$$$$49\!\cdots\!34$$$$T^{10} +$$$$21\!\cdots\!80$$$$T^{11} +$$$$35\!\cdots\!41$$$$T^{12} )^{2}$$
$23$ $$1 -$$$$48\!\cdots\!40$$$$T^{2} +$$$$11\!\cdots\!14$$$$T^{4} -$$$$18\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!95$$$$T^{8} -$$$$16\!\cdots\!80$$$$T^{10} +$$$$10\!\cdots\!60$$$$T^{12} -$$$$50\!\cdots\!80$$$$T^{14} +$$$$19\!\cdots\!95$$$$T^{16} -$$$$53\!\cdots\!80$$$$T^{18} +$$$$10\!\cdots\!74$$$$T^{20} -$$$$13\!\cdots\!40$$$$T^{22} +$$$$88\!\cdots\!61$$$$T^{24}$$
$29$ $$1 -$$$$22\!\cdots\!72$$$$T^{2} +$$$$28\!\cdots\!86$$$$T^{4} -$$$$24\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!95$$$$T^{8} -$$$$79\!\cdots\!92$$$$T^{10} +$$$$32\!\cdots\!44$$$$T^{12} -$$$$11\!\cdots\!32$$$$T^{14} +$$$$29\!\cdots\!95$$$$T^{16} -$$$$64\!\cdots\!20$$$$T^{18} +$$$$10\!\cdots\!66$$$$T^{20} -$$$$11\!\cdots\!72$$$$T^{22} +$$$$71\!\cdots\!21$$$$T^{24}$$
$31$ $$( 1 -$$$$31\!\cdots\!92$$$$T +$$$$14\!\cdots\!06$$$$T^{2} -$$$$17\!\cdots\!20$$$$T^{3} +$$$$45\!\cdots\!95$$$$T^{4} +$$$$33\!\cdots\!08$$$$T^{5} +$$$$56\!\cdots\!84$$$$T^{6} +$$$$15\!\cdots\!28$$$$T^{7} +$$$$99\!\cdots\!95$$$$T^{8} -$$$$17\!\cdots\!20$$$$T^{9} +$$$$69\!\cdots\!66$$$$T^{10} -$$$$71\!\cdots\!92$$$$T^{11} +$$$$10\!\cdots\!41$$$$T^{12} )^{2}$$
$37$ $$( 1 +$$$$52\!\cdots\!76$$$$T +$$$$22\!\cdots\!58$$$$T^{2} +$$$$91\!\cdots\!04$$$$T^{3} +$$$$20\!\cdots\!59$$$$T^{4} +$$$$67\!\cdots\!68$$$$T^{5} +$$$$10\!\cdots\!44$$$$T^{6} +$$$$26\!\cdots\!72$$$$T^{7} +$$$$31\!\cdots\!19$$$$T^{8} +$$$$54\!\cdots\!56$$$$T^{9} +$$$$51\!\cdots\!98$$$$T^{10} +$$$$47\!\cdots\!24$$$$T^{11} +$$$$35\!\cdots\!21$$$$T^{12} )^{2}$$
$41$ $$1 -$$$$69\!\cdots\!12$$$$T^{2} +$$$$25\!\cdots\!06$$$$T^{4} -$$$$62\!\cdots\!20$$$$T^{6} +$$$$13\!\cdots\!95$$$$T^{8} -$$$$30\!\cdots\!92$$$$T^{10} +$$$$66\!\cdots\!24$$$$T^{12} -$$$$11\!\cdots\!72$$$$T^{14} +$$$$19\!\cdots\!95$$$$T^{16} -$$$$32\!\cdots\!20$$$$T^{18} +$$$$49\!\cdots\!66$$$$T^{20} -$$$$50\!\cdots\!12$$$$T^{22} +$$$$26\!\cdots\!41$$$$T^{24}$$
$43$ $$( 1 -$$$$50\!\cdots\!96$$$$T +$$$$42\!\cdots\!98$$$$T^{2} -$$$$15\!\cdots\!04$$$$T^{3} +$$$$97\!\cdots\!99$$$$T^{4} -$$$$27\!\cdots\!08$$$$T^{5} +$$$$13\!\cdots\!04$$$$T^{6} -$$$$32\!\cdots\!92$$$$T^{7} +$$$$13\!\cdots\!99$$$$T^{8} -$$$$25\!\cdots\!96$$$$T^{9} +$$$$83\!\cdots\!98$$$$T^{10} -$$$$11\!\cdots\!04$$$$T^{11} +$$$$26\!\cdots\!01$$$$T^{12} )^{2}$$
$47$ $$1 -$$$$20\!\cdots\!60$$$$T^{2} +$$$$22\!\cdots\!94$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!35$$$$T^{8} -$$$$46\!\cdots\!20$$$$T^{10} +$$$$17\!\cdots\!00$$$$T^{12} -$$$$56\!\cdots\!20$$$$T^{14} +$$$$14\!\cdots\!35$$$$T^{16} -$$$$29\!\cdots\!20$$$$T^{18} +$$$$46\!\cdots\!14$$$$T^{20} -$$$$51\!\cdots\!60$$$$T^{22} +$$$$29\!\cdots\!21$$$$T^{24}$$
$53$ $$1 -$$$$21\!\cdots\!60$$$$T^{2} +$$$$21\!\cdots\!94$$$$T^{4} -$$$$14\!\cdots\!20$$$$T^{6} +$$$$66\!\cdots\!35$$$$T^{8} -$$$$26\!\cdots\!20$$$$T^{10} +$$$$90\!\cdots\!00$$$$T^{12} -$$$$28\!\cdots\!20$$$$T^{14} +$$$$82\!\cdots\!35$$$$T^{16} -$$$$19\!\cdots\!20$$$$T^{18} +$$$$32\!\cdots\!14$$$$T^{20} -$$$$35\!\cdots\!60$$$$T^{22} +$$$$18\!\cdots\!21$$$$T^{24}$$
$59$ $$1 -$$$$11\!\cdots\!12$$$$T^{2} +$$$$61\!\cdots\!06$$$$T^{4} -$$$$24\!\cdots\!20$$$$T^{6} +$$$$72\!\cdots\!95$$$$T^{8} -$$$$18\!\cdots\!92$$$$T^{10} +$$$$38\!\cdots\!24$$$$T^{12} -$$$$69\!\cdots\!72$$$$T^{14} +$$$$10\!\cdots\!95$$$$T^{16} -$$$$13\!\cdots\!20$$$$T^{18} +$$$$13\!\cdots\!66$$$$T^{20} -$$$$92\!\cdots\!12$$$$T^{22} +$$$$32\!\cdots\!41$$$$T^{24}$$
$61$ $$( 1 -$$$$96\!\cdots\!12$$$$T +$$$$25\!\cdots\!46$$$$T^{2} -$$$$16\!\cdots\!20$$$$T^{3} +$$$$28\!\cdots\!95$$$$T^{4} -$$$$14\!\cdots\!92$$$$T^{5} +$$$$21\!\cdots\!24$$$$T^{6} -$$$$98\!\cdots\!52$$$$T^{7} +$$$$13\!\cdots\!95$$$$T^{8} -$$$$57\!\cdots\!20$$$$T^{9} +$$$$59\!\cdots\!66$$$$T^{10} -$$$$15\!\cdots\!12$$$$T^{11} +$$$$11\!\cdots\!81$$$$T^{12} )^{2}$$
$67$ $$( 1 +$$$$61\!\cdots\!76$$$$T +$$$$73\!\cdots\!18$$$$T^{2} -$$$$18\!\cdots\!16$$$$T^{3} +$$$$30\!\cdots\!19$$$$T^{4} -$$$$14\!\cdots\!12$$$$T^{5} +$$$$89\!\cdots\!84$$$$T^{6} -$$$$36\!\cdots\!08$$$$T^{7} +$$$$18\!\cdots\!39$$$$T^{8} -$$$$26\!\cdots\!64$$$$T^{9} +$$$$26\!\cdots\!98$$$$T^{10} +$$$$55\!\cdots\!24$$$$T^{11} +$$$$22\!\cdots\!41$$$$T^{12} )^{2}$$
$71$ $$1 -$$$$13\!\cdots\!72$$$$T^{2} +$$$$88\!\cdots\!86$$$$T^{4} -$$$$35\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!95$$$$T^{8} -$$$$25\!\cdots\!92$$$$T^{10} +$$$$58\!\cdots\!44$$$$T^{12} -$$$$12\!\cdots\!32$$$$T^{14} +$$$$25\!\cdots\!95$$$$T^{16} -$$$$43\!\cdots\!20$$$$T^{18} +$$$$53\!\cdots\!66$$$$T^{20} -$$$$41\!\cdots\!72$$$$T^{22} +$$$$14\!\cdots\!21$$$$T^{24}$$
$73$ $$( 1 -$$$$45\!\cdots\!36$$$$T +$$$$29\!\cdots\!38$$$$T^{2} -$$$$11\!\cdots\!84$$$$T^{3} +$$$$43\!\cdots\!39$$$$T^{4} -$$$$12\!\cdots\!08$$$$T^{5} +$$$$35\!\cdots\!64$$$$T^{6} -$$$$80\!\cdots\!52$$$$T^{7} +$$$$17\!\cdots\!79$$$$T^{8} -$$$$28\!\cdots\!56$$$$T^{9} +$$$$50\!\cdots\!98$$$$T^{10} -$$$$48\!\cdots\!64$$$$T^{11} +$$$$68\!\cdots\!81$$$$T^{12} )^{2}$$
$79$ $$( 1 -$$$$16\!\cdots\!60$$$$T +$$$$60\!\cdots\!54$$$$T^{2} -$$$$81\!\cdots\!20$$$$T^{3} +$$$$16\!\cdots\!35$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{5} +$$$$27\!\cdots\!00$$$$T^{6} -$$$$22\!\cdots\!20$$$$T^{7} +$$$$27\!\cdots\!35$$$$T^{8} -$$$$17\!\cdots\!20$$$$T^{9} +$$$$16\!\cdots\!14$$$$T^{10} -$$$$59\!\cdots\!60$$$$T^{11} +$$$$45\!\cdots\!61$$$$T^{12} )^{2}$$
$83$ $$1 -$$$$52\!\cdots\!60$$$$T^{2} +$$$$14\!\cdots\!34$$$$T^{4} -$$$$26\!\cdots\!20$$$$T^{6} +$$$$37\!\cdots\!35$$$$T^{8} -$$$$42\!\cdots\!20$$$$T^{10} +$$$$38\!\cdots\!00$$$$T^{12} -$$$$29\!\cdots\!20$$$$T^{14} +$$$$18\!\cdots\!35$$$$T^{16} -$$$$94\!\cdots\!20$$$$T^{18} +$$$$36\!\cdots\!14$$$$T^{20} -$$$$94\!\cdots\!60$$$$T^{22} +$$$$12\!\cdots\!81$$$$T^{24}$$
$89$ $$1 -$$$$10\!\cdots\!32$$$$T^{2} +$$$$50\!\cdots\!26$$$$T^{4} -$$$$16\!\cdots\!20$$$$T^{6} +$$$$36\!\cdots\!95$$$$T^{8} -$$$$63\!\cdots\!92$$$$T^{10} +$$$$85\!\cdots\!64$$$$T^{12} -$$$$90\!\cdots\!12$$$$T^{14} +$$$$74\!\cdots\!95$$$$T^{16} -$$$$46\!\cdots\!20$$$$T^{18} +$$$$20\!\cdots\!66$$$$T^{20} -$$$$60\!\cdots\!32$$$$T^{22} +$$$$83\!\cdots\!61$$$$T^{24}$$
$97$ $$( 1 -$$$$12\!\cdots\!64$$$$T +$$$$14\!\cdots\!78$$$$T^{2} -$$$$10\!\cdots\!96$$$$T^{3} +$$$$79\!\cdots\!79$$$$T^{4} -$$$$46\!\cdots\!12$$$$T^{5} +$$$$30\!\cdots\!24$$$$T^{6} -$$$$14\!\cdots\!68$$$$T^{7} +$$$$78\!\cdots\!59$$$$T^{8} -$$$$31\!\cdots\!24$$$$T^{9} +$$$$14\!\cdots\!98$$$$T^{10} -$$$$38\!\cdots\!36$$$$T^{11} +$$$$96\!\cdots\!61$$$$T^{12} )^{2}$$