# Properties

 Label 630.2.bv Level 630 Weight 2 Character orbit bv Rep. character $$\chi_{630}(73,\cdot)$$ Character field $$\Q(\zeta_{12})$$ Dimension 80 Newform subspaces 4 Sturm bound 288 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.bv (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$35$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$4$$ Sturm bound: $$288$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(630, [\chi])$$.

Total New Old
Modular forms 640 80 560
Cusp forms 512 80 432
Eisenstein series 128 0 128

## Trace form

 $$80q - 12q^{5} + O(q^{10})$$ $$80q - 12q^{5} + 12q^{10} + 4q^{11} + 40q^{16} + 36q^{17} + 8q^{22} + 12q^{23} - 4q^{25} + 12q^{26} + 12q^{28} + 24q^{31} + 32q^{35} + 20q^{37} + 24q^{38} + 24q^{43} - 8q^{46} + 12q^{47} + 32q^{50} + 12q^{53} - 12q^{56} - 24q^{58} + 84q^{61} - 24q^{65} + 36q^{68} - 36q^{70} + 48q^{71} - 60q^{73} - 16q^{77} - 12q^{80} + 48q^{82} - 104q^{85} + 4q^{86} + 4q^{88} - 32q^{91} - 24q^{92} - 44q^{95} - 56q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(630, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
630.2.bv.a $$16$$ $$5.031$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$-12$$ $$-8$$ $$q+\beta _{15}q^{2}+(\beta _{5}-\beta _{13})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$$
630.2.bv.b $$16$$ $$5.031$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$-12$$ $$-4$$ $$q-\beta _{6}q^{2}+(-\beta _{5}+\beta _{13})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots$$
630.2.bv.c $$16$$ $$5.031$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$12$$ $$8$$ $$q+\beta _{13}q^{2}+(\beta _{11}-\beta _{14})q^{4}+(\beta _{5}-\beta _{6}+\cdots)q^{5}+\cdots$$
630.2.bv.d $$32$$ $$5.031$$ None $$0$$ $$0$$ $$0$$ $$4$$

## Decomposition of $$S_{2}^{\mathrm{old}}(630, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(630, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - T^{4} + T^{8} )^{2}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)
$3$ 1
$5$ ($$1 + 12 T + 88 T^{2} + 472 T^{3} + 2033 T^{4} + 7336 T^{5} + 22744 T^{6} + 61572 T^{7} + 146544 T^{8} + 307860 T^{9} + 568600 T^{10} + 917000 T^{11} + 1270625 T^{12} + 1475000 T^{13} + 1375000 T^{14} + 937500 T^{15} + 390625 T^{16}$$)($$1 + 12 T + 64 T^{2} + 200 T^{3} + 398 T^{4} + 380 T^{5} - 992 T^{6} - 6636 T^{7} - 19341 T^{8} - 33180 T^{9} - 24800 T^{10} + 47500 T^{11} + 248750 T^{12} + 625000 T^{13} + 1000000 T^{14} + 937500 T^{15} + 390625 T^{16}$$)($$1 - 12 T + 66 T^{2} - 216 T^{3} + 450 T^{4} - 468 T^{5} - 864 T^{6} + 6252 T^{7} - 18241 T^{8} + 31260 T^{9} - 21600 T^{10} - 58500 T^{11} + 281250 T^{12} - 675000 T^{13} + 1031250 T^{14} - 937500 T^{15} + 390625 T^{16}$$)
$7$ ($$1 + 8 T + 6 T^{2} - 120 T^{3} - 275 T^{4} + 1120 T^{5} + 4770 T^{6} - 3216 T^{7} - 40872 T^{8} - 22512 T^{9} + 233730 T^{10} + 384160 T^{11} - 660275 T^{12} - 2016840 T^{13} + 705894 T^{14} + 6588344 T^{15} + 5764801 T^{16}$$)($$1 + 4 T + 34 T^{2} + 80 T^{3} + 445 T^{4} + 704 T^{5} + 3926 T^{6} + 5044 T^{7} + 29688 T^{8} + 35308 T^{9} + 192374 T^{10} + 241472 T^{11} + 1068445 T^{12} + 1344560 T^{13} + 4000066 T^{14} + 3294172 T^{15} + 5764801 T^{16}$$)($$1 - 8 T + 32 T^{2} - 80 T^{3} + 73 T^{4} + 160 T^{5} - 416 T^{6} - 696 T^{7} + 4944 T^{8} - 4872 T^{9} - 20384 T^{10} + 54880 T^{11} + 175273 T^{12} - 1344560 T^{13} + 3764768 T^{14} - 6588344 T^{15} + 5764801 T^{16}$$)
$11$ ($$1 + 4 T - 42 T^{2} - 72 T^{3} + 1354 T^{4} + 52 T^{5} - 25972 T^{6} + 39092 T^{7} + 336125 T^{8} - 1014028 T^{9} - 2401652 T^{10} + 15137644 T^{11} - 1718882 T^{12} - 141984000 T^{13} + 309093362 T^{14} + 600162532 T^{15} - 4715588236 T^{16} + 6601787852 T^{17} + 37400296802 T^{18} - 188980704000 T^{19} - 25166151362 T^{20} + 2437932703844 T^{21} - 4254673018772 T^{22} - 19760537034788 T^{23} + 72051378876125 T^{24} + 92176891136572 T^{25} - 673646791737172 T^{26} + 14836206871772 T^{27} + 4249432022080234 T^{28} - 2485635274363032 T^{29} - 15949493010496122 T^{30} + 16708992677662604 T^{31} + 45949729863572161 T^{32}$$)($$1 + 4 T - 42 T^{2} - 72 T^{3} + 1354 T^{4} + 52 T^{5} - 25972 T^{6} + 39092 T^{7} + 336125 T^{8} - 1014028 T^{9} - 2401652 T^{10} + 15137644 T^{11} - 1718882 T^{12} - 141984000 T^{13} + 309093362 T^{14} + 600162532 T^{15} - 4715588236 T^{16} + 6601787852 T^{17} + 37400296802 T^{18} - 188980704000 T^{19} - 25166151362 T^{20} + 2437932703844 T^{21} - 4254673018772 T^{22} - 19760537034788 T^{23} + 72051378876125 T^{24} + 92176891136572 T^{25} - 673646791737172 T^{26} + 14836206871772 T^{27} + 4249432022080234 T^{28} - 2485635274363032 T^{29} - 15949493010496122 T^{30} + 16708992677662604 T^{31} + 45949729863572161 T^{32}$$)($$( 1 - 6 T + 5 T^{2} + 18 T^{3} - 29 T^{4} + 252 T^{5} - 544 T^{6} + 516 T^{7} - 6650 T^{8} + 5676 T^{9} - 65824 T^{10} + 335412 T^{11} - 424589 T^{12} + 2898918 T^{13} + 8857805 T^{14} - 116923026 T^{15} + 214358881 T^{16} )^{2}$$)
$13$ ($$1 + 16 T + 128 T^{2} + 744 T^{3} + 3314 T^{4} + 10768 T^{5} + 24864 T^{6} + 28864 T^{7} - 4655 T^{8} + 572216 T^{9} + 6801728 T^{10} + 43610328 T^{11} + 198012786 T^{12} + 641713232 T^{13} + 1520861600 T^{14} + 2393411960 T^{15} + 3297590084 T^{16} + 31114355480 T^{17} + 257025610400 T^{18} + 1409843970704 T^{19} + 5655443180946 T^{20} + 16192209514104 T^{21} + 32830641925952 T^{22} + 35905705403672 T^{23} - 3797226506255 T^{24} + 306088269902272 T^{25} + 3427713541333536 T^{26} + 19297983122990416 T^{27} + 77209854095902034 T^{28} + 225339079304636232 T^{29} + 503984177369508992 T^{30} + 818974288225452112 T^{31} + 665416609183179841 T^{32}$$)($$1 - 16 T + 128 T^{2} - 744 T^{3} + 3314 T^{4} - 10768 T^{5} + 24864 T^{6} - 28864 T^{7} - 4655 T^{8} - 572216 T^{9} + 6801728 T^{10} - 43610328 T^{11} + 198012786 T^{12} - 641713232 T^{13} + 1520861600 T^{14} - 2393411960 T^{15} + 3297590084 T^{16} - 31114355480 T^{17} + 257025610400 T^{18} - 1409843970704 T^{19} + 5655443180946 T^{20} - 16192209514104 T^{21} + 32830641925952 T^{22} - 35905705403672 T^{23} - 3797226506255 T^{24} - 306088269902272 T^{25} + 3427713541333536 T^{26} - 19297983122990416 T^{27} + 77209854095902034 T^{28} - 225339079304636232 T^{29} + 503984177369508992 T^{30} - 818974288225452112 T^{31} + 665416609183179841 T^{32}$$)($$1 + 714 T^{4} + 266977 T^{8} + 68653746 T^{12} + 13237862628 T^{16} + 1960819639506 T^{20} + 217781340700417 T^{24} + 16634832777451434 T^{28} + 665416609183179841 T^{32}$$)
$17$ ($$1 - 12 T + 144 T^{2} - 968 T^{3} + 6368 T^{4} - 26396 T^{5} + 107888 T^{6} - 145332 T^{7} - 233838 T^{8} + 8508352 T^{9} - 40945088 T^{10} + 232534236 T^{11} - 555402720 T^{12} + 1622229852 T^{13} + 5913536576 T^{14} - 33762615104 T^{15} + 261999162867 T^{16} - 573964456768 T^{17} + 1709012070464 T^{18} + 7970015262876 T^{19} - 46387790577120 T^{20} + 330165362724252 T^{21} - 988314886811072 T^{22} + 3491305869096896 T^{23} - 1631197168488558 T^{24} - 17234613267062004 T^{25} + 217501549931641712 T^{26} - 904640974936280668 T^{27} + 3710138406679118048 T^{28} - 9587631535852947016 T^{29} + 24246407024553733776 T^{30} - 34349076618117789516 T^{31} + 48661191875666868481 T^{32}$$)($$1 + 12 T - 760 T^{3} - 4000 T^{4} + 9644 T^{5} + 164912 T^{6} + 417780 T^{7} - 2033646 T^{8} - 15406336 T^{9} - 21694928 T^{10} + 141136884 T^{11} + 768607008 T^{12} + 1405601268 T^{13} - 1667112976 T^{14} - 24205669888 T^{15} - 123458993805 T^{16} - 411496388096 T^{17} - 481795650064 T^{18} + 6905719029684 T^{19} + 64194825915168 T^{20} + 200394192705588 T^{21} - 523662821550032 T^{22} - 6321815470032128 T^{23} - 14186221216859886 T^{24} + 49543643042916660 T^{25} + 332461586110845488 T^{26} + 330518167990812652 T^{27} - 2330488948919044000 T^{28} - 7527479305008512120 T^{29} + 34349076618117789516 T^{31} + 48661191875666868481 T^{32}$$)($$1 - 36 T + 648 T^{2} - 7776 T^{3} + 70080 T^{4} - 507420 T^{5} + 3088368 T^{6} - 16357284 T^{7} + 77768530 T^{8} - 343436904 T^{9} + 1468660248 T^{10} - 6351048828 T^{11} + 28493424864 T^{12} - 131692528476 T^{13} + 608521516920 T^{14} - 2723900163624 T^{15} + 11569567164915 T^{16} - 46306302781608 T^{17} + 175862718389880 T^{18} - 647005392402588 T^{19} + 2379799338066144 T^{20} - 9017581135777596 T^{21} + 35449888073657112 T^{22} - 140925443446588392 T^{23} + 542494401823131730 T^{24} - 1939775574818354148 T^{25} + 6226131050341877232 T^{26} - 17390245624419136860 T^{27} + 40830166385061650880 T^{28} - 77017998783876566112 T^{29} +$$$$10\!\cdots\!92$$$$T^{30} -$$$$10\!\cdots\!48$$$$T^{31} + 48661191875666868481 T^{32}$$)
$19$ ($$1 + 8 T - 22 T^{2} + 144 T^{3} + 3371 T^{4} - 4000 T^{5} - 7722 T^{6} + 652040 T^{7} - 370487 T^{8} - 2996624 T^{9} + 89400308 T^{10} - 59920656 T^{11} - 371517522 T^{12} + 10373644960 T^{13} - 8723924296 T^{14} - 39539357168 T^{15} + 977996314130 T^{16} - 751247786192 T^{17} - 3149336670856 T^{18} + 71152830780640 T^{19} - 48416534984562 T^{20} - 148369476400944 T^{21} + 4205916251531348 T^{22} - 2678597506009136 T^{23} - 6292189320370967 T^{24} + 210405286459819160 T^{25} - 47344093642739322 T^{26} - 465961035592876000 T^{27} + 7461084592172028731 T^{28} + 6055629618565016496 T^{29} - 17578147087223450662 T^{30} +$$$$12\!\cdots\!92$$$$T^{31} +$$$$28\!\cdots\!81$$$$T^{32}$$)($$1 - 8 T - 22 T^{2} - 144 T^{3} + 3371 T^{4} + 4000 T^{5} - 7722 T^{6} - 652040 T^{7} - 370487 T^{8} + 2996624 T^{9} + 89400308 T^{10} + 59920656 T^{11} - 371517522 T^{12} - 10373644960 T^{13} - 8723924296 T^{14} + 39539357168 T^{15} + 977996314130 T^{16} + 751247786192 T^{17} - 3149336670856 T^{18} - 71152830780640 T^{19} - 48416534984562 T^{20} + 148369476400944 T^{21} + 4205916251531348 T^{22} + 2678597506009136 T^{23} - 6292189320370967 T^{24} - 210405286459819160 T^{25} - 47344093642739322 T^{26} + 465961035592876000 T^{27} + 7461084592172028731 T^{28} - 6055629618565016496 T^{29} - 17578147087223450662 T^{30} -$$$$12\!\cdots\!92$$$$T^{31} +$$$$28\!\cdots\!81$$$$T^{32}$$)($$1 - 90 T^{2} + 4011 T^{4} - 123414 T^{6} + 3034057 T^{8} - 61998180 T^{10} + 1094493006 T^{12} - 18780034728 T^{14} + 344074548210 T^{16} - 6779592536808 T^{18} + 142635423034926 T^{20} - 2916758998496580 T^{22} + 51529098329487337 T^{24} - 756659411140252614 T^{26} + 8877606140374371771 T^{28} - 71910601720459570890 T^{30} +$$$$28\!\cdots\!81$$$$T^{32}$$)
$23$ ($$1 + 32 T + 536 T^{2} + 6152 T^{3} + 54743 T^{4} + 405160 T^{5} + 2601288 T^{6} + 14731984 T^{7} + 73350129 T^{8} + 313225808 T^{9} + 1063218752 T^{10} + 1942762160 T^{11} - 9374200334 T^{12} - 140082082224 T^{13} - 1076056625056 T^{14} - 6480812802528 T^{15} - 33308810016754 T^{16} - 149058694458144 T^{17} - 569233954654624 T^{18} - 1704378694419408 T^{19} - 2623285595666894 T^{20} + 12504283629180880 T^{21} + 157394533153790528 T^{22} + 1066479201735536176 T^{23} + 5744120872478451249 T^{24} + 26534552190230332592 T^{25} +$$$$10\!\cdots\!12$$$$T^{26} +$$$$38\!\cdots\!20$$$$T^{27} +$$$$11\!\cdots\!03$$$$T^{28} +$$$$31\!\cdots\!16$$$$T^{29} +$$$$62\!\cdots\!24$$$$T^{30} +$$$$85\!\cdots\!24$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)($$1 - 40 T + 776 T^{2} - 9784 T^{3} + 91799 T^{4} - 705368 T^{5} + 4787160 T^{6} - 29946728 T^{7} + 172895985 T^{8} - 902021104 T^{9} + 4178360768 T^{10} - 17091640336 T^{11} + 61289409778 T^{12} - 182133009408 T^{13} + 342869112800 T^{14} + 372347426016 T^{15} - 5577275049586 T^{16} + 8563990798368 T^{17} + 181377760671200 T^{18} - 2216012325467136 T^{19} + 17151289721685298 T^{20} - 110007659635131248 T^{21} + 618547350853602752 T^{22} - 3071224408630233488 T^{23} + 13539654936478996785 T^{24} - 53938628839308543064 T^{25} +$$$$19\!\cdots\!40$$$$T^{26} -$$$$67\!\cdots\!36$$$$T^{27} +$$$$20\!\cdots\!79$$$$T^{28} -$$$$49\!\cdots\!72$$$$T^{29} +$$$$89\!\cdots\!84$$$$T^{30} -$$$$10\!\cdots\!80$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)($$1 - 4 T + 8 T^{2} + 128 T^{3} - 198 T^{4} + 292 T^{5} + 8608 T^{6} - 3564 T^{7} - 17335 T^{8} - 1017836 T^{9} + 5500848 T^{10} + 425132 T^{11} + 20115594 T^{12} - 929862264 T^{13} + 4560471720 T^{14} + 14572270428 T^{15} - 102219015884 T^{16} + 335162219844 T^{17} + 2412489539880 T^{18} - 11313634166088 T^{19} + 5629167940554 T^{20} + 2736295372276 T^{21} + 814322923933872 T^{22} - 3465553913672692 T^{23} - 1357520929846135 T^{24} - 6419308085454132 T^{25} + 356599408527090592 T^{26} + 278220449310866684 T^{27} - 4339095637540023558 T^{28} + 64516654327867825024 T^{29} + 92742690596309998472 T^{30} -$$$$10\!\cdots\!28$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$1 - 248 T^{2} + 31418 T^{4} - 2677416 T^{6} + 171374865 T^{8} - 8752284864 T^{10} + 370628380738 T^{12} - 13351487849248 T^{14} + 415501762604420 T^{16} - 11228601281217568 T^{18} + 262138411756753378 T^{20} - 5206063149142513344 T^{22} + 85729661487925625265 T^{24} -$$$$11\!\cdots\!16$$$$T^{26} +$$$$11\!\cdots\!38$$$$T^{28} -$$$$73\!\cdots\!88$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$)($$1 - 248 T^{2} + 31418 T^{4} - 2677416 T^{6} + 171374865 T^{8} - 8752284864 T^{10} + 370628380738 T^{12} - 13351487849248 T^{14} + 415501762604420 T^{16} - 11228601281217568 T^{18} + 262138411756753378 T^{20} - 5206063149142513344 T^{22} + 85729661487925625265 T^{24} -$$$$11\!\cdots\!16$$$$T^{26} +$$$$11\!\cdots\!38$$$$T^{28} -$$$$73\!\cdots\!88$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$)($$( 1 - 70 T^{2} + 2685 T^{4} - 80906 T^{6} + 2505752 T^{8} - 68041946 T^{10} + 1899049485 T^{12} - 41637632470 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$1 + 24 T + 334 T^{2} + 3408 T^{3} + 27899 T^{4} + 199680 T^{5} + 1273586 T^{6} + 7589688 T^{7} + 43727785 T^{8} + 250780944 T^{9} + 1474786764 T^{10} + 8526426768 T^{11} + 48339918958 T^{12} + 259935770400 T^{13} + 1349761682520 T^{14} + 7075762017840 T^{15} + 37942723845666 T^{16} + 219348622553040 T^{17} + 1297120976901720 T^{18} + 7743746535986400 T^{19} + 44642930296011118 T^{20} + 244104359431513968 T^{21} + 1308878681740078284 T^{22} + 6899639338664300784 T^{23} + 37295035913646998185 T^{24} +$$$$20\!\cdots\!48$$$$T^{25} +$$$$10\!\cdots\!86$$$$T^{26} +$$$$50\!\cdots\!80$$$$T^{27} +$$$$21\!\cdots\!39$$$$T^{28} +$$$$83\!\cdots\!28$$$$T^{29} +$$$$25\!\cdots\!14$$$$T^{30} +$$$$56\!\cdots\!24$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$)($$1 + 24 T + 334 T^{2} + 3408 T^{3} + 27899 T^{4} + 199680 T^{5} + 1273586 T^{6} + 7589688 T^{7} + 43727785 T^{8} + 250780944 T^{9} + 1474786764 T^{10} + 8526426768 T^{11} + 48339918958 T^{12} + 259935770400 T^{13} + 1349761682520 T^{14} + 7075762017840 T^{15} + 37942723845666 T^{16} + 219348622553040 T^{17} + 1297120976901720 T^{18} + 7743746535986400 T^{19} + 44642930296011118 T^{20} + 244104359431513968 T^{21} + 1308878681740078284 T^{22} + 6899639338664300784 T^{23} + 37295035913646998185 T^{24} +$$$$20\!\cdots\!48$$$$T^{25} +$$$$10\!\cdots\!86$$$$T^{26} +$$$$50\!\cdots\!80$$$$T^{27} +$$$$21\!\cdots\!39$$$$T^{28} +$$$$83\!\cdots\!28$$$$T^{29} +$$$$25\!\cdots\!14$$$$T^{30} +$$$$56\!\cdots\!24$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$)($$( 1 - 12 T + 156 T^{2} - 1296 T^{3} + 10686 T^{4} - 72996 T^{5} + 489600 T^{6} - 2919012 T^{7} + 17046947 T^{8} - 90489372 T^{9} + 470505600 T^{10} - 2174623836 T^{11} + 9868745406 T^{12} - 37103379696 T^{13} + 138450574236 T^{14} - 330151369332 T^{15} + 852891037441 T^{16} )^{2}$$)
$37$ ($$1 + 8 T - 184 T^{2} - 1704 T^{3} + 16375 T^{4} + 186288 T^{5} - 883880 T^{6} - 13446752 T^{7} + 28690305 T^{8} + 694378184 T^{9} - 355698912 T^{10} - 26449712536 T^{11} - 17643340702 T^{12} + 717223895656 T^{13} + 1384593306720 T^{14} - 9602300166392 T^{15} - 59171912839138 T^{16} - 355285106156504 T^{17} + 1895508236899680 T^{18} + 36329541986663368 T^{19} - 33066461053401022 T^{20} - 1834127728758744952 T^{21} - 912626092170967008 T^{22} + 65918624447323666472 T^{23} +$$$$10\!\cdots\!05$$$$T^{24} -$$$$17\!\cdots\!04$$$$T^{25} -$$$$42\!\cdots\!20$$$$T^{26} +$$$$33\!\cdots\!44$$$$T^{27} +$$$$10\!\cdots\!75$$$$T^{28} -$$$$41\!\cdots\!88$$$$T^{29} -$$$$16\!\cdots\!76$$$$T^{30} +$$$$26\!\cdots\!44$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$)($$1 + 8 T + 248 T^{2} + 1752 T^{3} + 30199 T^{4} + 206976 T^{5} + 2540200 T^{6} + 17745904 T^{7} + 169033089 T^{8} + 1209867128 T^{9} + 9485737632 T^{10} + 68571970520 T^{11} + 465855584354 T^{12} + 3322795052152 T^{13} + 20345020109280 T^{14} + 139940970795928 T^{15} + 794780046531614 T^{16} + 5177815919449336 T^{17} + 27852332529604320 T^{18} + 168309537776655256 T^{19} + 873088367828476994 T^{20} + 4755051775144147640 T^{21} + 24337807551267523488 T^{22} +$$$$11\!\cdots\!24$$$$T^{23} +$$$$59\!\cdots\!69$$$$T^{24} +$$$$23\!\cdots\!08$$$$T^{25} +$$$$12\!\cdots\!00$$$$T^{26} +$$$$36\!\cdots\!88$$$$T^{27} +$$$$19\!\cdots\!19$$$$T^{28} +$$$$42\!\cdots\!44$$$$T^{29} +$$$$22\!\cdots\!72$$$$T^{30} +$$$$26\!\cdots\!44$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$)($$1 - 4 T + 8 T^{2} + 240 T^{3} - 3049 T^{4} + 13544 T^{5} - 984 T^{6} - 514972 T^{7} + 3800209 T^{8} - 12256568 T^{9} - 16248928 T^{10} + 295990824 T^{11} + 119383314 T^{12} - 6433305488 T^{13} + 16607500784 T^{14} + 478842888760 T^{15} - 3828375485890 T^{16} + 17717186884120 T^{17} + 22735668573296 T^{18} - 325866222883664 T^{19} + 223743551149554 T^{20} + 20525174971850568 T^{21} - 41690303687539552 T^{22} - 1163539007448259544 T^{23} + 13348156033105669489 T^{24} - 66926657065750392844 T^{25} - 4731647022459163416 T^{26} +$$$$24\!\cdots\!72$$$$T^{27} -$$$$20\!\cdots\!69$$$$T^{28} +$$$$58\!\cdots\!80$$$$T^{29} +$$$$72\!\cdots\!12$$$$T^{30} -$$$$13\!\cdots\!72$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$)
$41$ ($$1 - 172 T^{2} + 13974 T^{4} - 791464 T^{6} + 43993793 T^{8} - 2592908552 T^{10} + 134337801798 T^{12} - 5711936189732 T^{14} + 227906401150852 T^{16} - 9601764734939492 T^{18} + 379606521146518278 T^{20} - 12316585909380369032 T^{22} +$$$$35\!\cdots\!53$$$$T^{24} -$$$$10\!\cdots\!64$$$$T^{26} +$$$$31\!\cdots\!94$$$$T^{28} -$$$$65\!\cdots\!92$$$$T^{30} +$$$$63\!\cdots\!41$$$$T^{32}$$)($$1 - 172 T^{2} + 13974 T^{4} - 791464 T^{6} + 43993793 T^{8} - 2592908552 T^{10} + 134337801798 T^{12} - 5711936189732 T^{14} + 227906401150852 T^{16} - 9601764734939492 T^{18} + 379606521146518278 T^{20} - 12316585909380369032 T^{22} +$$$$35\!\cdots\!53$$$$T^{24} -$$$$10\!\cdots\!64$$$$T^{26} +$$$$31\!\cdots\!94$$$$T^{28} -$$$$65\!\cdots\!92$$$$T^{30} +$$$$63\!\cdots\!41$$$$T^{32}$$)($$( 1 - 188 T^{2} + 17802 T^{4} - 1146544 T^{6} + 54447827 T^{8} - 1927340464 T^{10} + 50304197322 T^{12} - 893019597308 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$1 + 24 T + 288 T^{2} + 2456 T^{3} + 18144 T^{4} + 133752 T^{5} + 1000544 T^{6} + 6828856 T^{7} + 39963644 T^{8} + 227672888 T^{9} + 1517218592 T^{10} + 11117129912 T^{11} + 81274307360 T^{12} + 589957036568 T^{13} + 4310182316384 T^{14} + 31623166948824 T^{15} + 218467749390534 T^{16} + 1359796178799432 T^{17} + 7969527102994016 T^{18} + 46905714106411976 T^{19} + 277860683276675360 T^{20} + 1634311958991847016 T^{21} + 9590889544724607008 T^{22} + 61885728202879567016 T^{23} +$$$$46\!\cdots\!44$$$$T^{24} +$$$$34\!\cdots\!08$$$$T^{25} +$$$$21\!\cdots\!56$$$$T^{26} +$$$$12\!\cdots\!64$$$$T^{27} +$$$$72\!\cdots\!44$$$$T^{28} +$$$$42\!\cdots\!08$$$$T^{29} +$$$$21\!\cdots\!12$$$$T^{30} +$$$$76\!\cdots\!68$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$)($$1 + 24 T + 288 T^{2} + 2456 T^{3} + 18144 T^{4} + 133752 T^{5} + 1000544 T^{6} + 6828856 T^{7} + 39963644 T^{8} + 227672888 T^{9} + 1517218592 T^{10} + 11117129912 T^{11} + 81274307360 T^{12} + 589957036568 T^{13} + 4310182316384 T^{14} + 31623166948824 T^{15} + 218467749390534 T^{16} + 1359796178799432 T^{17} + 7969527102994016 T^{18} + 46905714106411976 T^{19} + 277860683276675360 T^{20} + 1634311958991847016 T^{21} + 9590889544724607008 T^{22} + 61885728202879567016 T^{23} +$$$$46\!\cdots\!44$$$$T^{24} +$$$$34\!\cdots\!08$$$$T^{25} +$$$$21\!\cdots\!56$$$$T^{26} +$$$$12\!\cdots\!64$$$$T^{27} +$$$$72\!\cdots\!44$$$$T^{28} +$$$$42\!\cdots\!08$$$$T^{29} +$$$$21\!\cdots\!12$$$$T^{30} +$$$$76\!\cdots\!68$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$)($$( 1 + 4 T + 8 T^{2} + 32 T^{3} + 3425 T^{4} + 20696 T^{5} + 55896 T^{6} + 651228 T^{7} + 6527152 T^{8} + 28002804 T^{9} + 103351704 T^{10} + 1645476872 T^{11} + 11709393425 T^{12} + 4704270176 T^{13} + 50570904392 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2}$$)
$47$ ($$1 - 24 T + 216 T^{2} + 40 T^{3} - 18457 T^{4} + 150040 T^{5} + 44552 T^{6} - 9386184 T^{7} + 61817361 T^{8} + 96146848 T^{9} - 3977203520 T^{10} + 21778214496 T^{11} + 50868959058 T^{12} - 1343760123600 T^{13} + 6047140835168 T^{14} + 23428375060624 T^{15} - 394415861681970 T^{16} + 1101133627849328 T^{17} + 13358134104886112 T^{18} - 139513207312522800 T^{19} + 248224293005100498 T^{20} + 4994724756032621472 T^{21} - 42871133149336758080 T^{22} + 48710216156441750624 T^{23} +$$$$14\!\cdots\!21$$$$T^{24} -$$$$10\!\cdots\!28$$$$T^{25} +$$$$23\!\cdots\!48$$$$T^{26} +$$$$37\!\cdots\!20$$$$T^{27} -$$$$21\!\cdots\!37$$$$T^{28} +$$$$21\!\cdots\!80$$$$T^{29} +$$$$55\!\cdots\!04$$$$T^{30} -$$$$28\!\cdots\!32$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$)($$1 + 72 T^{2} - 616 T^{3} + 3239 T^{4} - 50872 T^{5} + 298520 T^{6} - 2523744 T^{7} + 15099153 T^{8} - 113209408 T^{9} + 814540096 T^{10} - 3210267840 T^{11} + 24971489874 T^{12} - 159883078560 T^{13} + 379015320032 T^{14} - 3041257505584 T^{15} + 39450787699278 T^{16} - 142939102762448 T^{17} + 837244841950688 T^{18} - 16599540865334880 T^{19} + 121852904679850194 T^{20} - 736258900236674880 T^{21} + 8780103088888331584 T^{22} - 57354503546728915904 T^{23} +$$$$35\!\cdots\!33$$$$T^{24} -$$$$28\!\cdots\!48$$$$T^{25} +$$$$15\!\cdots\!80$$$$T^{26} -$$$$12\!\cdots\!16$$$$T^{27} +$$$$37\!\cdots\!99$$$$T^{28} -$$$$33\!\cdots\!32$$$$T^{29} +$$$$18\!\cdots\!68$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$)($$1 + 12 T + 72 T^{2} + 288 T^{3} - 449 T^{4} - 2880 T^{5} + 39240 T^{6} + 500820 T^{7} + 44673 T^{8} - 37644384 T^{9} - 288946944 T^{10} - 1723474464 T^{11} + 9984376994 T^{12} + 179746348152 T^{13} + 1044056404368 T^{14} + 4096130163936 T^{15} + 8350203627806 T^{16} + 192518117704992 T^{17} + 2306320597248912 T^{18} + 18661805104185096 T^{19} + 48720574714458914 T^{20} - 395270263010401248 T^{21} - 3114621328032504576 T^{22} - 19071515289987429792 T^{23} + 1063721609040849153 T^{24} +$$$$56\!\cdots\!40$$$$T^{25} +$$$$20\!\cdots\!60$$$$T^{26} -$$$$71\!\cdots\!40$$$$T^{27} -$$$$52\!\cdots\!09$$$$T^{28} +$$$$15\!\cdots\!76$$$$T^{29} +$$$$18\!\cdots\!68$$$$T^{30} +$$$$14\!\cdots\!16$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$1 + 44 T + 1016 T^{2} + 15712 T^{3} + 178130 T^{4} + 1540980 T^{5} + 10390336 T^{6} + 58117132 T^{7} + 357423825 T^{8} + 3614994372 T^{9} + 45716027504 T^{10} + 505985826004 T^{11} + 4503741642258 T^{12} + 32039845594664 T^{13} + 184020453561240 T^{14} + 916952178658276 T^{15} + 5330817007952772 T^{16} + 48598465468888628 T^{17} + 516913454053523160 T^{18} + 4769996092596792328 T^{19} + 35536687857145546098 T^{20} +$$$$21\!\cdots\!72$$$$T^{21} +$$$$10\!\cdots\!16$$$$T^{22} +$$$$42\!\cdots\!64$$$$T^{23} +$$$$22\!\cdots\!25$$$$T^{24} +$$$$19\!\cdots\!56$$$$T^{25} +$$$$18\!\cdots\!64$$$$T^{26} +$$$$14\!\cdots\!60$$$$T^{27} +$$$$87\!\cdots\!30$$$$T^{28} +$$$$40\!\cdots\!76$$$$T^{29} +$$$$14\!\cdots\!04$$$$T^{30} +$$$$32\!\cdots\!08$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$)($$1 - 28 T + 344 T^{2} - 2912 T^{3} + 25298 T^{4} - 232644 T^{5} + 1917568 T^{6} - 14158268 T^{7} + 104375121 T^{8} - 770687892 T^{9} + 5270701040 T^{10} - 39047642852 T^{11} + 318443399442 T^{12} - 2209044273928 T^{13} + 14320519756920 T^{14} - 111315852463604 T^{15} + 873920983850628 T^{16} - 5899740180571012 T^{17} + 40226339997188280 T^{18} - 328875884369578856 T^{19} + 2512671592872511602 T^{20} - 16329548252980066036 T^{21} +$$$$11\!\cdots\!60$$$$T^{22} -$$$$90\!\cdots\!04$$$$T^{23} +$$$$64\!\cdots\!81$$$$T^{24} -$$$$46\!\cdots\!44$$$$T^{25} +$$$$33\!\cdots\!32$$$$T^{26} -$$$$21\!\cdots\!68$$$$T^{27} +$$$$12\!\cdots\!18$$$$T^{28} -$$$$75\!\cdots\!76$$$$T^{29} +$$$$47\!\cdots\!36$$$$T^{30} -$$$$20\!\cdots\!96$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$)($$1 - 28 T + 392 T^{2} - 1808 T^{3} - 25321 T^{4} + 567704 T^{5} - 4335448 T^{6} - 1435172 T^{7} + 392677969 T^{8} - 4105212392 T^{9} + 14977011104 T^{10} + 124625519288 T^{11} - 2189877391278 T^{12} + 14281594759536 T^{13} - 14623477702416 T^{14} - 648224117593688 T^{15} + 7103959219709822 T^{16} - 34355878232465464 T^{17} - 41077348866086544 T^{18} + 2126200983015441072 T^{19} - 17279185948208624718 T^{20} + 52117830479026168984 T^{21} +$$$$33\!\cdots\!16$$$$T^{22} -$$$$48\!\cdots\!04$$$$T^{23} +$$$$24\!\cdots\!09$$$$T^{24} -$$$$47\!\cdots\!76$$$$T^{25} -$$$$75\!\cdots\!52$$$$T^{26} +$$$$52\!\cdots\!88$$$$T^{27} -$$$$12\!\cdots\!61$$$$T^{28} -$$$$47\!\cdots\!84$$$$T^{29} +$$$$54\!\cdots\!48$$$$T^{30} -$$$$20\!\cdots\!96$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$)
$59$ ($$1 + 8 T - 236 T^{2} - 1632 T^{3} + 30563 T^{4} + 163648 T^{5} - 2682716 T^{6} - 9836488 T^{7} + 178469613 T^{8} + 332320192 T^{9} - 10946443072 T^{10} - 3817987168 T^{11} + 756189977838 T^{12} - 175777223888 T^{13} - 56160273686696 T^{14} + 6297587928448 T^{15} + 3675723794639094 T^{16} + 371557687778432 T^{17} - 195493912703388776 T^{18} - 36100950464893552 T^{19} + 9163026946045045518 T^{20} - 2729571799673395232 T^{21} -$$$$46\!\cdots\!52$$$$T^{22} +$$$$82\!\cdots\!48$$$$T^{23} +$$$$26\!\cdots\!73$$$$T^{24} -$$$$85\!\cdots\!32$$$$T^{25} -$$$$13\!\cdots\!16$$$$T^{26} +$$$$49\!\cdots\!32$$$$T^{27} +$$$$54\!\cdots\!03$$$$T^{28} -$$$$17\!\cdots\!28$$$$T^{29} -$$$$14\!\cdots\!96$$$$T^{30} +$$$$29\!\cdots\!92$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$)($$1 - 8 T - 236 T^{2} + 1632 T^{3} + 30563 T^{4} - 163648 T^{5} - 2682716 T^{6} + 9836488 T^{7} + 178469613 T^{8} - 332320192 T^{9} - 10946443072 T^{10} + 3817987168 T^{11} + 756189977838 T^{12} + 175777223888 T^{13} - 56160273686696 T^{14} - 6297587928448 T^{15} + 3675723794639094 T^{16} - 371557687778432 T^{17} - 195493912703388776 T^{18} + 36100950464893552 T^{19} + 9163026946045045518 T^{20} + 2729571799673395232 T^{21} -$$$$46\!\cdots\!52$$$$T^{22} -$$$$82\!\cdots\!48$$$$T^{23} +$$$$26\!\cdots\!73$$$$T^{24} +$$$$85\!\cdots\!32$$$$T^{25} -$$$$13\!\cdots\!16$$$$T^{26} -$$$$49\!\cdots\!32$$$$T^{27} +$$$$54\!\cdots\!03$$$$T^{28} +$$$$17\!\cdots\!28$$$$T^{29} -$$$$14\!\cdots\!96$$$$T^{30} -$$$$29\!\cdots\!92$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$)($$1 - 320 T^{2} + 54244 T^{4} - 6209920 T^{6} + 532238186 T^{8} - 36082424000 T^{10} + 2045329177232 T^{12} - 105925693186240 T^{14} + 5808829694261683 T^{16} - 368727337981301440 T^{18} + 24783992004353124752 T^{20} -$$$$15\!\cdots\!00$$$$T^{22} +$$$$78\!\cdots\!06$$$$T^{24} -$$$$31\!\cdots\!20$$$$T^{26} +$$$$96\!\cdots\!64$$$$T^{28} -$$$$19\!\cdots\!20$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$1 - 24 T + 468 T^{2} - 6624 T^{3} + 79480 T^{4} - 829512 T^{5} + 7576872 T^{6} - 61686168 T^{7} + 432435906 T^{8} - 2460066384 T^{9} + 8493304812 T^{10} + 36652670904 T^{11} - 1139116635712 T^{12} + 15272951044440 T^{13} - 160222534806948 T^{14} + 1465410452960880 T^{15} - 12014486118540493 T^{16} + 89390037630613680 T^{17} - 596188052016653508 T^{18} + 3466669701018035640 T^{19} - 15772027818523273792 T^{20} + 30956710267288726104 T^{21} +$$$$43\!\cdots\!32$$$$T^{22} -$$$$77\!\cdots\!64$$$$T^{23} +$$$$82\!\cdots\!86$$$$T^{24} -$$$$72\!\cdots\!88$$$$T^{25} +$$$$54\!\cdots\!72$$$$T^{26} -$$$$36\!\cdots\!32$$$$T^{27} +$$$$21\!\cdots\!80$$$$T^{28} -$$$$10\!\cdots\!44$$$$T^{29} +$$$$46\!\cdots\!88$$$$T^{30} -$$$$14\!\cdots\!24$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$)($$1 - 24 T + 468 T^{2} - 6624 T^{3} + 79480 T^{4} - 829512 T^{5} + 7576872 T^{6} - 61686168 T^{7} + 432435906 T^{8} - 2460066384 T^{9} + 8493304812 T^{10} + 36652670904 T^{11} - 1139116635712 T^{12} + 15272951044440 T^{13} - 160222534806948 T^{14} + 1465410452960880 T^{15} - 12014486118540493 T^{16} + 89390037630613680 T^{17} - 596188052016653508 T^{18} + 3466669701018035640 T^{19} - 15772027818523273792 T^{20} + 30956710267288726104 T^{21} +$$$$43\!\cdots\!32$$$$T^{22} -$$$$77\!\cdots\!64$$$$T^{23} +$$$$82\!\cdots\!86$$$$T^{24} -$$$$72\!\cdots\!88$$$$T^{25} +$$$$54\!\cdots\!72$$$$T^{26} -$$$$36\!\cdots\!32$$$$T^{27} +$$$$21\!\cdots\!80$$$$T^{28} -$$$$10\!\cdots\!44$$$$T^{29} +$$$$46\!\cdots\!88$$$$T^{30} -$$$$14\!\cdots\!24$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$)($$( 1 + 6 T + 207 T^{2} + 1170 T^{3} + 23079 T^{4} + 119796 T^{5} + 1969284 T^{6} + 9244740 T^{7} + 135493478 T^{8} + 563929140 T^{9} + 7327705764 T^{10} + 27191415876 T^{11} + 319548164439 T^{12} + 988177672170 T^{13} + 10664717492727 T^{14} + 18856457016126 T^{15} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$1 - 36 T + 960 T^{2} - 19808 T^{3} + 351456 T^{4} - 5525148 T^{5} + 79071728 T^{6} - 1045655452 T^{7} + 12919151378 T^{8} - 150131218232 T^{9} + 1651372846640 T^{10} - 17248158507668 T^{11} + 171756254361824 T^{12} - 1633413949285436 T^{13} + 14868335084466800 T^{14} - 129646495999196328 T^{15} + 1083758100261010803 T^{16} - 8686315231946153976 T^{17} + 66743956194171465200 T^{18} -$$$$49\!\cdots\!68$$$$T^{19} +$$$$34\!\cdots\!04$$$$T^{20} -$$$$23\!\cdots\!76$$$$T^{21} +$$$$14\!\cdots\!60$$$$T^{22} -$$$$90\!\cdots\!36$$$$T^{23} +$$$$52\!\cdots\!98$$$$T^{24} -$$$$28\!\cdots\!44$$$$T^{25} +$$$$14\!\cdots\!72$$$$T^{26} -$$$$67\!\cdots\!84$$$$T^{27} +$$$$28\!\cdots\!16$$$$T^{28} -$$$$10\!\cdots\!96$$$$T^{29} +$$$$35\!\cdots\!40$$$$T^{30} -$$$$88\!\cdots\!48$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$)($$1 + 84 T + 3216 T^{2} + 71776 T^{3} + 957024 T^{4} + 5741436 T^{5} - 40997200 T^{6} - 1158667636 T^{7} - 7828301806 T^{8} + 48210145144 T^{9} + 1314996417536 T^{10} + 8000602781236 T^{11} - 45952509042976 T^{12} - 1085658642853028 T^{13} - 5561139754106368 T^{14} + 34677137718483048 T^{15} + 641528682100762995 T^{16} + 2323368227138364216 T^{17} - 24963956356183485952 T^{18} -$$$$32\!\cdots\!64$$$$T^{19} -$$$$92\!\cdots\!96$$$$T^{20} +$$$$10\!\cdots\!52$$$$T^{21} +$$$$11\!\cdots\!84$$$$T^{22} +$$$$29\!\cdots\!12$$$$T^{23} -$$$$31\!\cdots\!46$$$$T^{24} -$$$$31\!\cdots\!92$$$$T^{25} -$$$$74\!\cdots\!00$$$$T^{26} +$$$$70\!\cdots\!88$$$$T^{27} +$$$$78\!\cdots\!64$$$$T^{28} +$$$$39\!\cdots\!12$$$$T^{29} +$$$$11\!\cdots\!64$$$$T^{30} +$$$$20\!\cdots\!12$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$)($$1 - 32 T + 512 T^{2} - 3896 T^{3} - 10509 T^{4} + 663104 T^{5} - 8249312 T^{6} + 44834240 T^{7} + 165963557 T^{8} - 5505231928 T^{9} + 49681319072 T^{10} - 159136747544 T^{11} - 1412717145882 T^{12} + 21627811976512 T^{13} - 103082665870656 T^{14} - 335467607852040 T^{15} + 7552610469365206 T^{16} - 22476329726086680 T^{17} - 462738087093374784 T^{18} + 6504845613491678656 T^{19} - 28467834145442833722 T^{20} -$$$$21\!\cdots\!08$$$$T^{21} +$$$$44\!\cdots\!68$$$$T^{22} -$$$$33\!\cdots\!44$$$$T^{23} +$$$$67\!\cdots\!37$$$$T^{24} +$$$$12\!\cdots\!80$$$$T^{25} -$$$$15\!\cdots\!88$$$$T^{26} +$$$$80\!\cdots\!32$$$$T^{27} -$$$$85\!\cdots\!49$$$$T^{28} -$$$$21\!\cdots\!52$$$$T^{29} +$$$$18\!\cdots\!48$$$$T^{30} -$$$$78\!\cdots\!76$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$)
$71$ ($$( 1 - 16 T + 392 T^{2} - 4432 T^{3} + 72412 T^{4} - 697872 T^{5} + 8835896 T^{6} - 71051216 T^{7} + 737387398 T^{8} - 5044636336 T^{9} + 44541751736 T^{10} - 249776065392 T^{11} + 1840110644572 T^{12} - 7996344483632 T^{13} + 50215311297032 T^{14} - 145521922534256 T^{15} + 645753531245761 T^{16} )^{2}$$)($$( 1 - 16 T + 392 T^{2} - 4432 T^{3} + 72412 T^{4} - 697872 T^{5} + 8835896 T^{6} - 71051216 T^{7} + 737387398 T^{8} - 5044636336 T^{9} + 44541751736 T^{10} - 249776065392 T^{11} + 1840110644572 T^{12} - 7996344483632 T^{13} + 50215311297032 T^{14} - 145521922534256 T^{15} + 645753531245761 T^{16} )^{2}$$)($$( 1 + 4 T + 94 T^{2} - 964 T^{3} - 1158 T^{4} - 68444 T^{5} + 473854 T^{6} + 1431644 T^{7} + 25411681 T^{8} )^{4}$$)
$73$ ($$1 + 40 T + 512 T^{2} - 1704 T^{3} - 115340 T^{4} - 995264 T^{5} + 4224032 T^{6} + 134702512 T^{7} + 706330218 T^{8} - 4851409768 T^{9} - 89219066272 T^{10} - 434676875784 T^{11} + 2356262737872 T^{12} + 56320191577336 T^{13} + 325034384414112 T^{14} - 1853571681237960 T^{15} - 38795165571506349 T^{16} - 135310732730371080 T^{17} + 1732108234542802848 T^{18} + 21909511966840518712 T^{19} + 66913717089408883152 T^{20} -$$$$90\!\cdots\!12$$$$T^{21} -$$$$13\!\cdots\!08$$$$T^{22} -$$$$53\!\cdots\!96$$$$T^{23} +$$$$56\!\cdots\!58$$$$T^{24} +$$$$79\!\cdots\!56$$$$T^{25} +$$$$18\!\cdots\!68$$$$T^{26} -$$$$31\!\cdots\!28$$$$T^{27} -$$$$26\!\cdots\!40$$$$T^{28} -$$$$28\!\cdots\!32$$$$T^{29} +$$$$62\!\cdots\!08$$$$T^{30} +$$$$35\!\cdots\!80$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$)($$1 - 16 T + 416 T^{2} - 4440 T^{3} + 68404 T^{4} - 584824 T^{5} + 7229216 T^{6} - 53644504 T^{7} + 544685226 T^{8} - 2736110216 T^{9} + 18336160640 T^{10} + 72833715600 T^{11} - 1347659375280 T^{12} + 28574375923760 T^{13} - 284065502536896 T^{14} + 3320804755103832 T^{15} - 26931854637922413 T^{16} + 242418747122579736 T^{17} - 1513785063019118784 T^{18} + 11115917998733343920 T^{19} - 38271155725110882480 T^{20} +$$$$15\!\cdots\!00$$$$T^{21} +$$$$27\!\cdots\!60$$$$T^{22} -$$$$30\!\cdots\!52$$$$T^{23} +$$$$43\!\cdots\!06$$$$T^{24} -$$$$31\!\cdots\!52$$$$T^{25} +$$$$31\!\cdots\!84$$$$T^{26} -$$$$18\!\cdots\!48$$$$T^{27} +$$$$15\!\cdots\!84$$$$T^{28} -$$$$74\!\cdots\!20$$$$T^{29} +$$$$50\!\cdots\!44$$$$T^{30} -$$$$14\!\cdots\!12$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$)($$1 + 12 T + 72 T^{2} + 288 T^{3} + 12064 T^{4} + 83700 T^{5} + 177264 T^{6} - 3613524 T^{7} + 19255890 T^{8} - 96865032 T^{9} - 2225167272 T^{10} - 39441388620 T^{11} + 6056513120 T^{12} - 1050679100940 T^{13} - 2551121523144 T^{14} - 71385024146568 T^{15} + 865631164724531 T^{16} - 5211106762699464 T^{17} - 13594926596834376 T^{18} - 408732031810375980 T^{19} + 171994319201421920 T^{20} - 81764822336595471660 T^{21} -$$$$33\!\cdots\!08$$$$T^{22} -$$$$10\!\cdots\!04$$$$T^{23} +$$$$15\!\cdots\!90$$$$T^{24} -$$$$21\!\cdots\!12$$$$T^{25} +$$$$76\!\cdots\!36$$$$T^{26} +$$$$26\!\cdots\!00$$$$T^{27} +$$$$27\!\cdots\!44$$$$T^{28} +$$$$48\!\cdots\!04$$$$T^{29} +$$$$87\!\cdots\!48$$$$T^{30} +$$$$10\!\cdots\!84$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$)
$79$ ($$1 - 12 T + 518 T^{2} - 5640 T^{3} + 137639 T^{4} - 1484376 T^{5} + 26147402 T^{6} - 279843540 T^{7} + 3950719429 T^{8} - 41115674928 T^{9} + 499791767516 T^{10} - 4982987471808 T^{11} + 54520483633690 T^{12} - 515559074594184 T^{13} + 5198103978036912 T^{14} - 46390749479828928 T^{15} + 436708099187124858 T^{16} - 3664869208906485312 T^{17} + 32441366926928367792 T^{18} -$$$$25\!\cdots\!76$$$$T^{19} +$$$$21\!\cdots\!90$$$$T^{20} -$$$$15\!\cdots\!92$$$$T^{21} +$$$$12\!\cdots\!36$$$$T^{22} -$$$$78\!\cdots\!52$$$$T^{23} +$$$$59\!\cdots\!69$$$$T^{24} -$$$$33\!\cdots\!60$$$$T^{25} +$$$$24\!\cdots\!02$$$$T^{26} -$$$$11\!\cdots\!04$$$$T^{27} +$$$$81\!\cdots\!99$$$$T^{28} -$$$$26\!\cdots\!60$$$$T^{29} +$$$$19\!\cdots\!58$$$$T^{30} -$$$$34\!\cdots\!88$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$)($$1 + 12 T + 518 T^{2} + 5640 T^{3} + 137639 T^{4} + 1484376 T^{5} + 26147402 T^{6} + 279843540 T^{7} + 3950719429 T^{8} + 41115674928 T^{9} + 499791767516 T^{10} + 4982987471808 T^{11} + 54520483633690 T^{12} + 515559074594184 T^{13} + 5198103978036912 T^{14} + 46390749479828928 T^{15} + 436708099187124858 T^{16} + 3664869208906485312 T^{17} + 32441366926928367792 T^{18} +$$$$25\!\cdots\!76$$$$T^{19} +$$$$21\!\cdots\!90$$$$T^{20} +$$$$15\!\cdots\!92$$$$T^{21} +$$$$12\!\cdots\!36$$$$T^{22} +$$$$78\!\cdots\!52$$$$T^{23} +$$$$59\!\cdots\!69$$$$T^{24} +$$$$33\!\cdots\!60$$$$T^{25} +$$$$24\!\cdots\!02$$$$T^{26} +$$$$11\!\cdots\!04$$$$T^{27} +$$$$81\!\cdots\!99$$$$T^{28} +$$$$26\!\cdots\!60$$$$T^{29} +$$$$19\!\cdots\!58$$$$T^{30} +$$$$34\!\cdots\!88$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$)($$1 + 344 T^{2} + 56836 T^{4} + 6614224 T^{6} + 630756554 T^{8} + 45125337896 T^{10} + 2052833339792 T^{12} + 54520165812136 T^{14} + 1660868031271123 T^{16} + 340260354833540776 T^{18} + 79958024864398923152 T^{20} +$$$$10\!\cdots\!16$$$$T^{22} +$$$$95\!\cdots\!94$$$$T^{24} +$$$$62\!\cdots\!24$$$$T^{26} +$$$$33\!\cdots\!76$$$$T^{28} +$$$$12\!\cdots\!64$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$)
$83$ ($$1 - 16 T + 128 T^{2} - 48 T^{3} - 16022 T^{4} + 55056 T^{5} + 1171072 T^{6} - 29664624 T^{7} + 191234977 T^{8} + 1519529248 T^{9} - 27088145920 T^{10} + 210706506976 T^{11} + 1209312464722 T^{12} - 24316714882016 T^{13} + 89859490185984 T^{14} + 1324590918805248 T^{15} - 28693493560058396 T^{16} + 109941046260835584 T^{17} + 619042027891243776 T^{18} - 13903981452243282592 T^{19} + 57391939140077851762 T^{20} +$$$$82\!\cdots\!68$$$$T^{21} -$$$$88\!\cdots\!80$$$$T^{22} +$$$$41\!\cdots\!96$$$$T^{23} +$$$$43\!\cdots\!57$$$$T^{24} -$$$$55\!\cdots\!72$$$$T^{25} +$$$$18\!\cdots\!28$$$$T^{26} +$$$$70\!\cdots\!52$$$$T^{27} -$$$$17\!\cdots\!42$$$$T^{28} -$$$$42\!\cdots\!24$$$$T^{29} +$$$$94\!\cdots\!12$$$$T^{30} -$$$$97\!\cdots\!12$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$)($$1 + 16 T + 128 T^{2} + 48 T^{3} - 16022 T^{4} - 55056 T^{5} + 1171072 T^{6} + 29664624 T^{7} + 191234977 T^{8} - 1519529248 T^{9} - 27088145920 T^{10} - 210706506976 T^{11} + 1209312464722 T^{12} + 24316714882016 T^{13} + 89859490185984 T^{14} - 1324590918805248 T^{15} - 28693493560058396 T^{16} - 109941046260835584 T^{17} + 619042027891243776 T^{18} + 13903981452243282592 T^{19} + 57391939140077851762 T^{20} -$$$$82\!\cdots\!68$$$$T^{21} -$$$$88\!\cdots\!80$$$$T^{22} -$$$$41\!\cdots\!96$$$$T^{23} +$$$$43\!\cdots\!57$$$$T^{24} +$$$$55\!\cdots\!72$$$$T^{25} +$$$$18\!\cdots\!28$$$$T^{26} -$$$$70\!\cdots\!52$$$$T^{27} -$$$$17\!\cdots\!42$$$$T^{28} +$$$$42\!\cdots\!24$$$$T^{29} +$$$$94\!\cdots\!12$$$$T^{30} +$$$$97\!\cdots\!12$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$)($$1 - 25638 T^{4} + 318852241 T^{8} - 2957648820318 T^{12} + 22747905510477444 T^{16} -$$$$14\!\cdots\!78$$$$T^{20} +$$$$71\!\cdots\!81$$$$T^{24} -$$$$27\!\cdots\!18$$$$T^{28} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$1 - 16 T - 440 T^{2} + 6624 T^{3} + 131636 T^{4} - 1629968 T^{5} - 29449232 T^{6} + 280328720 T^{7} + 5329291050 T^{8} - 36595065152 T^{9} - 800017753480 T^{10} + 3635275214288 T^{11} + 102057968083920 T^{12} - 258547916350832 T^{13} - 11209896420041672 T^{14} + 8808353629704256 T^{15} + 1069443829226298003 T^{16} + 783943473043678784 T^{17} - 88793589543150083912 T^{18} -$$$$18\!\cdots\!08$$$$T^{19} +$$$$64\!\cdots\!20$$$$T^{20} +$$$$20\!\cdots\!12$$$$T^{21} -$$$$39\!\cdots\!80$$$$T^{22} -$$$$16\!\cdots\!08$$$$T^{23} +$$$$20\!\cdots\!50$$$$T^{24} +$$$$98\!\cdots\!80$$$$T^{25} -$$$$91\!\cdots\!32$$$$T^{26} -$$$$45\!\cdots\!52$$$$T^{27} +$$$$32\!\cdots\!56$$$$T^{28} +$$$$14\!\cdots\!56$$$$T^{29} -$$$$86\!\cdots\!40$$$$T^{30} -$$$$27\!\cdots\!84$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$)($$1 + 16 T - 440 T^{2} - 6624 T^{3} + 131636 T^{4} + 1629968 T^{5} - 29449232 T^{6} - 280328720 T^{7} + 5329291050 T^{8} + 36595065152 T^{9} - 800017753480 T^{10} - 3635275214288 T^{11} + 102057968083920 T^{12} + 258547916350832 T^{13} - 11209896420041672 T^{14} - 8808353629704256 T^{15} + 1069443829226298003 T^{16} - 783943473043678784 T^{17} - 88793589543150083912 T^{18} +$$$$18\!\cdots\!08$$$$T^{19} +$$$$64\!\cdots\!20$$$$T^{20} -$$$$20\!\cdots\!12$$$$T^{21} -$$$$39\!\cdots\!80$$$$T^{22} +$$$$16\!\cdots\!08$$$$T^{23} +$$$$20\!\cdots\!50$$$$T^{24} -$$$$98\!\cdots\!80$$$$T^{25} -$$$$91\!\cdots\!32$$$$T^{26} +$$$$45\!\cdots\!52$$$$T^{27} +$$$$32\!\cdots\!56$$$$T^{28} -$$$$14\!\cdots\!56$$$$T^{29} -$$$$86\!\cdots\!40$$$$T^{30} +$$$$27\!\cdots\!84$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$)($$1 - 522 T^{2} + 144915 T^{4} - 27754278 T^{6} + 4091335849 T^{8} - 494840689332 T^{10} + 51768593923950 T^{12} - 4926383904774360 T^{14} + 445675116984224850 T^{16} - 39021886909717705560 T^{18} +$$$$32\!\cdots\!50$$$$T^{20} -$$$$24\!\cdots\!52$$$$T^{22} +$$$$16\!\cdots\!69$$$$T^{24} -$$$$86\!\cdots\!78$$$$T^{26} +$$$$35\!\cdots\!15$$$$T^{28} -$$$$10\!\cdots\!02$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$)
$97$ ($$1 - 44 T + 968 T^{2} - 15000 T^{3} + 231654 T^{4} - 3860388 T^{5} + 58116000 T^{6} - 743419588 T^{7} + 9111376513 T^{8} - 116138978204 T^{9} + 1430090287664 T^{10} - 15964265880436 T^{11} + 171574947065342 T^{12} - 1888109943974960 T^{13} + 20331243756190952 T^{14} - 203613182755137748 T^{15} + 1985416913924716484 T^{16} - 19750478727248361556 T^{17} +$$$$19\!\cdots\!68$$$$T^{18} -$$$$17\!\cdots\!80$$$$T^{19} +$$$$15\!\cdots\!02$$$$T^{20} -$$$$13\!\cdots\!52$$$$T^{21} +$$$$11\!\cdots\!56$$$$T^{22} -$$$$93\!\cdots\!52$$$$T^{23} +$$$$71\!\cdots\!93$$$$T^{24} -$$$$56\!\cdots\!96$$$$T^{25} +$$$$42\!\cdots\!00$$$$T^{26} -$$$$27\!\cdots\!64$$$$T^{27} +$$$$16\!\cdots\!14$$$$T^{28} -$$$$10\!\cdots\!00$$$$T^{29} +$$$$63\!\cdots\!92$$$$T^{30} -$$$$27\!\cdots\!92$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$)($$1 + 44 T + 968 T^{2} + 15000 T^{3} + 231654 T^{4} + 3860388 T^{5} + 58116000 T^{6} + 743419588 T^{7} + 9111376513 T^{8} + 116138978204 T^{9} + 1430090287664 T^{10} + 15964265880436 T^{11} + 171574947065342 T^{12} + 1888109943974960 T^{13} + 20331243756190952 T^{14} + 203613182755137748 T^{15} + 1985416913924716484 T^{16} + 19750478727248361556 T^{17} +$$$$19\!\cdots\!68$$$$T^{18} +$$$$17\!\cdots\!80$$$$T^{19} +$$$$15\!\cdots\!02$$$$T^{20} +$$$$13\!\cdots\!52$$$$T^{21} +$$$$11\!\cdots\!56$$$$T^{22} +$$$$93\!\cdots\!52$$$$T^{23} +$$$$71\!\cdots\!93$$$$T^{24} +$$$$56\!\cdots\!96$$$$T^{25} +$$$$42\!\cdots\!00$$$$T^{26} +$$$$27\!\cdots\!64$$$$T^{27} +$$$$16\!\cdots\!14$$$$T^{28} +$$$$10\!\cdots\!00$$$$T^{29} +$$$$63\!\cdots\!92$$$$T^{30} +$$$$27\!\cdots\!92$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$)($$( 1 + 3868 T^{4} + 90482118 T^{8} + 342431258908 T^{12} + 7837433594376961 T^{16} )^{2}$$)