# Properties

 Label 1008.2.ca Level 1008 Weight 2 Character orbit ca Rep. character $$\chi_{1008}(257,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 92 Newform subspaces 5 Sturm bound 384 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1008.ca (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$63$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$5$$ Sturm bound: $$384$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1008, [\chi])$$.

Total New Old
Modular forms 408 100 308
Cusp forms 360 92 268
Eisenstein series 48 8 40

## Trace form

 $$92q + 3q^{3} - 3q^{5} + q^{7} - q^{9} + O(q^{10})$$ $$92q + 3q^{3} - 3q^{5} + q^{7} - q^{9} + 3q^{11} + 7q^{15} + 6q^{19} - q^{21} + 3q^{23} - 37q^{25} + 6q^{29} - 3q^{33} - 15q^{35} - 2q^{37} + 10q^{39} + 8q^{43} - 15q^{45} + 6q^{47} - q^{49} + 9q^{51} - 2q^{57} + 6q^{59} - 40q^{63} + 2q^{67} - 9q^{69} - 6q^{73} + 57q^{75} - 27q^{77} + 2q^{79} + 15q^{81} - 30q^{83} + 3q^{85} + 3q^{87} - 12q^{89} + 12q^{91} + 19q^{93} + 43q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1008, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1008.2.ca.a $$2$$ $$8.049$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-3$$ $$-4$$ $$q+(-1+2\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}+\cdots$$
1008.2.ca.b $$10$$ $$8.049$$ 10.0.$$\cdots$$.1 None $$0$$ $$3$$ $$0$$ $$6$$ $$q+(-\beta _{1}+\beta _{7})q^{3}+(\beta _{1}-\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots$$
1008.2.ca.c $$16$$ $$8.049$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+(-\beta _{3}+\beta _{9})q^{3}+(-\beta _{9}-\beta _{13}+\beta _{14}+\cdots)q^{5}+\cdots$$
1008.2.ca.d $$16$$ $$8.049$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$1$$ $$q-\beta _{12}q^{3}+\beta _{9}q^{5}+(\beta _{2}-\beta _{15})q^{7}+\cdots$$
1008.2.ca.e $$48$$ $$8.049$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1008, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1008, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 3 T^{2}$$)($$1 - 3 T + 3 T^{2} - 9 T^{3} + 21 T^{4} - 27 T^{5} + 63 T^{6} - 81 T^{7} + 81 T^{8} - 243 T^{9} + 243 T^{10}$$)($$1 + 12 T^{3} + 9 T^{4} - 18 T^{5} + 81 T^{6} + 90 T^{7} - 126 T^{8} + 270 T^{9} + 729 T^{10} - 486 T^{11} + 729 T^{12} + 2916 T^{13} + 6561 T^{16}$$)($$1 - 3 T^{2} + 3 T^{3} + 9 T^{5} + 36 T^{6} - 18 T^{7} - 72 T^{8} - 54 T^{9} + 324 T^{10} + 243 T^{11} + 729 T^{13} - 2187 T^{14} + 6561 T^{16}$$)
$5$ ($$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$)($$1 - 16 T^{2} + 12 T^{3} + 129 T^{4} - 147 T^{5} - 726 T^{6} + 720 T^{7} + 3663 T^{8} - 1257 T^{9} - 18069 T^{10} - 6285 T^{11} + 91575 T^{12} + 90000 T^{13} - 453750 T^{14} - 459375 T^{15} + 2015625 T^{16} + 937500 T^{17} - 6250000 T^{18} + 9765625 T^{20}$$)($$1 - 16 T^{2} - 24 T^{3} + 123 T^{4} + 330 T^{5} - 98 T^{6} - 2124 T^{7} - 4594 T^{8} - 102 T^{9} + 38946 T^{10} + 84396 T^{11} - 70916 T^{12} - 641196 T^{13} - 735151 T^{14} + 1444464 T^{15} + 6891966 T^{16} + 7222320 T^{17} - 18378775 T^{18} - 80149500 T^{19} - 44322500 T^{20} + 263737500 T^{21} + 608531250 T^{22} - 7968750 T^{23} - 1794531250 T^{24} - 4148437500 T^{25} - 957031250 T^{26} + 16113281250 T^{27} + 30029296875 T^{28} - 29296875000 T^{29} - 97656250000 T^{30} + 152587890625 T^{32}$$)($$1 - 16 T^{2} - 24 T^{3} + 105 T^{4} + 357 T^{5} - 80 T^{6} - 2232 T^{7} - 2857 T^{8} + 6027 T^{9} + 16041 T^{10} + 777 T^{11} - 31496 T^{12} - 32724 T^{13} + 43664 T^{14} + 15327 T^{15} - 229716 T^{16} + 76635 T^{17} + 1091600 T^{18} - 4090500 T^{19} - 19685000 T^{20} + 2428125 T^{21} + 250640625 T^{22} + 470859375 T^{23} - 1116015625 T^{24} - 4359375000 T^{25} - 781250000 T^{26} + 17431640625 T^{27} + 25634765625 T^{28} - 29296875000 T^{29} - 97656250000 T^{30} + 152587890625 T^{32}$$)
$7$ ($$1 + 4 T + 7 T^{2}$$)($$1 - 6 T + 16 T^{2} - T^{3} - 122 T^{4} + 473 T^{5} - 854 T^{6} - 49 T^{7} + 5488 T^{8} - 14406 T^{9} + 16807 T^{10}$$)($$1 + 2 T + 6 T^{2} - 8 T^{3} + 23 T^{4} + 210 T^{5} + 463 T^{6} + 1148 T^{7} - 1350 T^{8} + 8036 T^{9} + 22687 T^{10} + 72030 T^{11} + 55223 T^{12} - 134456 T^{13} + 705894 T^{14} + 1647086 T^{15} + 5764801 T^{16}$$)($$1 - T + 3 T^{2} + 7 T^{3} - 82 T^{4} + 141 T^{5} - 20 T^{6} - 1111 T^{7} + 5328 T^{8} - 7777 T^{9} - 980 T^{10} + 48363 T^{11} - 196882 T^{12} + 117649 T^{13} + 352947 T^{14} - 823543 T^{15} + 5764801 T^{16}$$)
$11$ ($$1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 12 T + 105 T^{2} - 684 T^{3} + 3745 T^{4} - 17382 T^{5} + 72307 T^{6} - 270561 T^{7} + 952558 T^{8} - 3194589 T^{9} + 10673917 T^{10} - 35140479 T^{11} + 115259518 T^{12} - 360116691 T^{13} + 1058646787 T^{14} - 2799388482 T^{15} + 6634495945 T^{16} - 13329224964 T^{17} + 22507682505 T^{18} - 28295372292 T^{19} + 25937424601 T^{20}$$)($$1 + 12 T + 106 T^{2} + 696 T^{3} + 3735 T^{4} + 17808 T^{5} + 77432 T^{6} + 326142 T^{7} + 1351880 T^{8} + 5458080 T^{9} + 21295098 T^{10} + 79655676 T^{11} + 287397262 T^{12} + 1023482544 T^{13} + 3603683689 T^{14} + 12488798178 T^{15} + 42242848866 T^{16} + 137376779958 T^{17} + 436045726369 T^{18} + 1362255266064 T^{19} + 4207783312942 T^{20} + 12828626275476 T^{21} + 37725565107978 T^{22} + 106362538291680 T^{23} + 289787484046280 T^{24} + 769025775838122 T^{25} + 2008386661704632 T^{26} + 5080830230240688 T^{27} + 11722029987052935 T^{28} + 24027807652175976 T^{29} + 40253482359823546 T^{30} + 50126978032987812 T^{31} + 45949729863572161 T^{32}$$)($$1 - 6 T + 61 T^{2} - 294 T^{3} + 1701 T^{4} - 6204 T^{5} + 24845 T^{6} - 59553 T^{7} + 143666 T^{8} + 78393 T^{9} - 1097631 T^{10} + 8096325 T^{11} - 21925205 T^{12} + 68059401 T^{13} - 34454504 T^{14} - 43343730 T^{15} + 1352446632 T^{16} - 476781030 T^{17} - 4168994984 T^{18} + 90587062731 T^{19} - 321006926405 T^{20} + 1303921237575 T^{21} - 1944520271991 T^{22} + 1527657796203 T^{23} + 30796082997746 T^{24} - 140422858842123 T^{25} + 644415314211845 T^{26} - 1770073604470644 T^{27} + 5338466668802421 T^{28} - 10149677370315714 T^{29} + 23164739848577701 T^{30} - 25063489016493906 T^{31} + 45949729863572161 T^{32}$$)
$13$ ($$( 1 - 5 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$1 + 6 T + 68 T^{2} + 336 T^{3} + 2292 T^{4} + 9399 T^{5} + 51837 T^{6} + 187401 T^{7} + 909867 T^{8} + 3004662 T^{9} + 13054461 T^{10} + 39060606 T^{11} + 153767523 T^{12} + 411719997 T^{13} + 1480516557 T^{14} + 3489782907 T^{15} + 11063046228 T^{16} + 21083501712 T^{17} + 55469689028 T^{18} + 63626996238 T^{19} + 137858491849 T^{20}$$)($$1 - 6 T + 47 T^{2} - 210 T^{3} + 888 T^{4} - 2658 T^{5} + 9181 T^{6} - 12564 T^{7} + 21641 T^{8} + 214770 T^{9} - 1221396 T^{10} + 6263526 T^{11} - 19011605 T^{12} + 72302586 T^{13} - 179114935 T^{14} + 742674348 T^{15} - 2320143696 T^{16} + 9654766524 T^{17} - 30270424015 T^{18} + 158848781442 T^{19} - 542990450405 T^{20} + 2325603359118 T^{21} - 5895445205364 T^{22} + 13476498996090 T^{23} + 17653228533161 T^{24} - 133234930122372 T^{25} + 1265678813665669 T^{26} - 4763562327350346 T^{27} + 20688699588763128 T^{28} - 63603772384373130 T^{29} + 185056690127866583 T^{30} - 307115358084544542 T^{31} + 665416609183179841 T^{32}$$)($$1 + 3 T + 53 T^{2} + 150 T^{3} + 1317 T^{4} + 4452 T^{5} + 23056 T^{6} + 90522 T^{7} + 327950 T^{8} + 1218621 T^{9} + 4082037 T^{10} + 12551466 T^{11} + 42977824 T^{12} + 136810599 T^{13} + 317458436 T^{14} + 1828010007 T^{15} + 2359677564 T^{16} + 23764130091 T^{17} + 53650475684 T^{18} + 300572886003 T^{19} + 1227489631264 T^{20} + 4660271465538 T^{21} + 19703212929933 T^{22} + 76466660535057 T^{23} + 267518889951950 T^{24} + 959940492242706 T^{25} + 3178465388070544 T^{26} + 7978698074252724 T^{27} + 30683578106307477 T^{28} + 45431265988837950 T^{29} + 208680948442062317 T^{30} + 153557679042272271 T^{31} + 665416609183179841 T^{32}$$)
$17$ ($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 - 12 T + 26 T^{2} + 36 T^{3} + 1143 T^{4} - 5247 T^{5} - 21540 T^{6} + 73476 T^{7} + 337539 T^{8} + 599625 T^{9} - 13374333 T^{10} + 10193625 T^{11} + 97548771 T^{12} + 360987588 T^{13} - 1799042340 T^{14} - 7449989679 T^{15} + 27589241367 T^{16} + 14772192228 T^{17} + 181369693466 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20}$$)($$1 + 18 T + 95 T^{2} - 42 T^{3} - 849 T^{4} + 7584 T^{5} + 29152 T^{6} - 139356 T^{7} - 154873 T^{8} + 2564070 T^{9} - 13186653 T^{10} - 89037906 T^{11} + 222290986 T^{12} + 933503742 T^{13} - 5890959001 T^{14} - 7644891330 T^{15} + 98422426836 T^{16} - 129963152610 T^{17} - 1702487151289 T^{18} + 4586303884446 T^{19} + 18565965441706 T^{20} - 126421094099442 T^{21} - 318293746666557 T^{22} + 1052137081279110 T^{23} - 1080356482159993 T^{24} - 16525932117115932 T^{25} + 58770254185889248 T^{26} + 259918061597088672 T^{27} - 494646279408067089 T^{28} - 415992277382049354 T^{29} + 15995893523143088255 T^{30} + 51523614927176684274 T^{31} + 48661191875666868481 T^{32}$$)($$1 - 9 T - 31 T^{2} + 498 T^{3} + 6 T^{4} - 11613 T^{5} + 1180 T^{6} + 197118 T^{7} + 356723 T^{8} - 3738756 T^{9} - 13114257 T^{10} + 60977685 T^{11} + 321880693 T^{12} - 548795961 T^{13} - 7936687522 T^{14} + 1772823582 T^{15} + 160937528238 T^{16} + 30138000894 T^{17} - 2293702693858 T^{18} - 2696234556393 T^{19} + 26883797360053 T^{20} + 86579592891045 T^{21} - 316546283221233 T^{22} - 1534156175710788 T^{23} + 2488413121625843 T^{24} + 23375805039335646 T^{25} + 2378872802529820 T^{26} - 397999531820542029 T^{27} + 3495733423378566 T^{28} + 4932479860387156626 T^{29} - 5219712623341428799 T^{30} - 25761807463588342137 T^{31} + 48661191875666868481 T^{32}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} )$$)($$1 + 3 T + 71 T^{2} + 204 T^{3} + 2646 T^{4} + 8547 T^{5} + 74607 T^{6} + 258954 T^{7} + 1743726 T^{8} + 5989488 T^{9} + 35261703 T^{10} + 113800272 T^{11} + 629485086 T^{12} + 1776165486 T^{13} + 9722858847 T^{14} + 21163218153 T^{15} + 124483401126 T^{16} + 182349834756 T^{17} + 1205832975911 T^{18} + 968063093337 T^{19} + 6131066257801 T^{20}$$)($$1 + 80 T^{2} + 3483 T^{4} + 882 T^{5} + 107770 T^{6} + 26190 T^{7} + 2596658 T^{8} - 477594 T^{9} + 51326154 T^{10} - 58243698 T^{11} + 858124540 T^{12} - 2331197604 T^{13} + 13169269781 T^{14} - 60976736712 T^{15} + 222780924306 T^{16} - 1158557997528 T^{17} + 4754106390941 T^{18} - 15989684365836 T^{19} + 111831648177340 T^{20} - 144217162374102 T^{21} + 2414684133271674 T^{22} - 426907779315966 T^{23} + 44100504838916978 T^{24} + 8451190804832010 T^{25} + 660745010603213770 T^{26} + 102744408348229158 T^{27} + 7708975863107438763 T^{28} + 63920534862630729680 T^{30} +$$$$28\!\cdots\!81$$$$T^{32}$$)($$1 + 59 T^{2} + 1482 T^{4} - 279 T^{5} + 21358 T^{6} + 23166 T^{7} + 284075 T^{8} + 1924515 T^{9} + 8259087 T^{10} + 60795468 T^{11} + 210571285 T^{12} + 1125692802 T^{13} + 3051652292 T^{14} + 14070928500 T^{15} + 40349090466 T^{16} + 267347641500 T^{17} + 1101646477412 T^{18} + 7721126928918 T^{19} + 27441860432485 T^{20} + 150535597519332 T^{21} + 388556024170647 T^{22} + 1720269569781585 T^{23} + 4824605670872075 T^{24} + 7475383206748314 T^{25} + 130947313134113758 T^{26} - 32500782232603101 T^{27} + 3280132710056050602 T^{28} + 47141394461190163139 T^{30} +$$$$28\!\cdots\!81$$$$T^{32}$$)
$23$ ($$1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4}$$)($$1 - 15 T + 195 T^{2} - 1800 T^{3} + 15028 T^{4} - 107121 T^{5} + 712306 T^{6} - 4264599 T^{7} + 24178468 T^{8} - 125954103 T^{9} + 628044835 T^{10} - 2896944369 T^{11} + 12790409572 T^{12} - 51887376033 T^{13} + 199332423346 T^{14} - 689467498503 T^{15} + 2224683339892 T^{16} - 6128685804600 T^{17} + 15270642129795 T^{18} - 27017289921945 T^{19} + 41426511213649 T^{20}$$)($$1 - 6 T + 130 T^{2} - 708 T^{3} + 8559 T^{4} - 45012 T^{5} + 399452 T^{6} - 2115900 T^{7} + 15122540 T^{8} - 81277422 T^{9} + 486352566 T^{10} - 2645846004 T^{11} + 13727139058 T^{12} - 75456395706 T^{13} + 352786829089 T^{14} - 1928667059838 T^{15} + 8415100540206 T^{16} - 44359342376274 T^{17} + 186624232588081 T^{18} - 918077966554902 T^{19} + 3841416321129778 T^{20} - 17029572406923372 T^{21} + 71997634475241174 T^{22} - 276735434692157634 T^{23} + 1184261007351333740 T^{24} - 3811058916389561700 T^{25} + 16547902757314520348 T^{26} - 42887872823221682124 T^{27} +$$$$18\!\cdots\!39$$$$T^{28} -$$$$35\!\cdots\!64$$$$T^{29} +$$$$15\!\cdots\!70$$$$T^{30} -$$$$15\!\cdots\!42$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)($$1 + 21 T + 283 T^{2} + 2856 T^{3} + 23904 T^{4} + 177009 T^{5} + 1214978 T^{6} + 7994925 T^{7} + 50784536 T^{8} + 310700127 T^{9} + 1817756595 T^{10} + 10184685783 T^{11} + 55106115085 T^{12} + 290568927984 T^{13} + 1499561670325 T^{14} + 7546165977213 T^{15} + 36821082385998 T^{16} + 173561817475899 T^{17} + 793268123601925 T^{18} + 3535352146781328 T^{19} + 15420950351501485 T^{20} + 65552131046611569 T^{21} + 269093213526437955 T^{22} + 1057879698795731769 T^{23} + 3976987051198414616 T^{24} + 14400080441947075275 T^{25} + 50332299741336834722 T^{26} +$$$$16\!\cdots\!43$$$$T^{27} +$$$$52\!\cdots\!84$$$$T^{28} +$$$$14\!\cdots\!48$$$$T^{29} +$$$$32\!\cdots\!47$$$$T^{30} +$$$$55\!\cdots\!47$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$1 + 9 T + 56 T^{2} + 261 T^{3} + 841 T^{4}$$)($$1 + 15 T + 150 T^{2} + 1125 T^{3} + 6691 T^{4} + 30108 T^{5} + 81631 T^{6} - 232971 T^{7} - 5137202 T^{8} - 44535417 T^{9} - 275752187 T^{10} - 1291527093 T^{11} - 4320386882 T^{12} - 5681929719 T^{13} + 57736055311 T^{14} + 617549674092 T^{15} + 3979962840811 T^{16} + 19406110847625 T^{17} + 75036961944150 T^{18} + 217607189638035 T^{19} + 420707233300201 T^{20}$$)($$1 - 6 T + 196 T^{2} - 1104 T^{3} + 19989 T^{4} - 108858 T^{5} + 1435478 T^{6} - 7594740 T^{7} + 81322766 T^{8} - 416245488 T^{9} + 3844091772 T^{10} - 18872529852 T^{11} + 155981060614 T^{12} - 727660406082 T^{13} + 5509975202215 T^{14} - 24244157784798 T^{15} + 170557455776958 T^{16} - 703080575759142 T^{17} + 4633889145062815 T^{18} - 17746909643933898 T^{19} + 110322440532130534 T^{20} - 387097271801319948 T^{21} + 2286555434049814812 T^{22} - 7180183182179343792 T^{23} + 40681421983566770126 T^{24} -$$$$11\!\cdots\!60$$$$T^{25} +$$$$60\!\cdots\!78$$$$T^{26} -$$$$13\!\cdots\!82$$$$T^{27} +$$$$70\!\cdots\!49$$$$T^{28} -$$$$11\!\cdots\!56$$$$T^{29} +$$$$58\!\cdots\!76$$$$T^{30} -$$$$51\!\cdots\!94$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$)($$1 - 6 T + 106 T^{2} - 564 T^{3} + 5445 T^{4} - 23700 T^{5} + 153455 T^{6} - 499140 T^{7} + 2268674 T^{8} - 3839766 T^{9} + 25367247 T^{10} - 108756381 T^{11} + 2283543946 T^{12} - 15452150286 T^{13} + 152754700246 T^{14} - 806878620189 T^{15} + 5726182499856 T^{16} - 23399479985481 T^{17} + 128466702906886 T^{18} - 376862493325254 T^{19} + 1615107245670826 T^{20} - 2230718335391769 T^{21} + 15089030105167287 T^{22} - 66235488555503694 T^{23} + 1134896030677883714 T^{24} - 7241096842395252660 T^{25} + 64559628486082344455 T^{26} -$$$$28\!\cdots\!00$$$$T^{27} +$$$$19\!\cdots\!45$$$$T^{28} -$$$$57\!\cdots\!96$$$$T^{29} +$$$$31\!\cdots\!86$$$$T^{30} -$$$$51\!\cdots\!94$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$)
$31$ ($$1 - 50 T^{2} + 961 T^{4}$$)($$1 - 217 T^{2} + 22896 T^{4} - 1551294 T^{6} + 74760504 T^{8} - 2674186965 T^{10} + 71844844344 T^{12} - 1432652586174 T^{14} + 20320284280176 T^{16} - 185077355124697 T^{18} + 819628286980801 T^{20}$$)($$1 - 292 T^{2} + 41022 T^{4} - 3694910 T^{6} + 240356405 T^{8} - 12117919608 T^{10} + 500572972675 T^{12} - 17863857948586 T^{14} + 576631021225626 T^{16} - 17167167488591146 T^{18} + 462289652297788675 T^{20} - 10754698258162077048 T^{22} +$$$$20\!\cdots\!05$$$$T^{24} -$$$$30\!\cdots\!10$$$$T^{26} +$$$$32\!\cdots\!42$$$$T^{28} -$$$$22\!\cdots\!32$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$)($$1 - 220 T^{2} + 25434 T^{4} - 2035931 T^{6} + 125854031 T^{8} - 6377263569 T^{10} + 274749236767 T^{12} - 10292358079372 T^{14} + 339402697885782 T^{16} - 9890956114276492 T^{18} + 253736689888296607 T^{20} - 5659844892194697489 T^{22} +$$$$10\!\cdots\!71$$$$T^{24} -$$$$16\!\cdots\!31$$$$T^{26} +$$$$20\!\cdots\!74$$$$T^{28} -$$$$16\!\cdots\!20$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$)
$37$ ($$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$)($$1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 405021462 T^{11} + 14702124973 T^{12} - 27012839676 T^{13} - 606842086834 T^{14} + 246587111092 T^{15} + 18419349890211 T^{16} + 18416784163802 T^{17} - 340710507030337 T^{18} - 779770438770462 T^{19} + 4808584372417849 T^{20}$$)($$1 + 2 T - 116 T^{2} - 656 T^{3} + 3854 T^{4} + 48538 T^{5} + 98460 T^{6} - 747702 T^{7} - 5251023 T^{8} - 50514234 T^{9} - 417738792 T^{10} + 801423570 T^{11} + 29966075382 T^{12} + 116541321036 T^{13} - 322717550496 T^{14} - 3343790405826 T^{15} - 14327838321804 T^{16} - 123720245015562 T^{17} - 441800326629024 T^{18} + 5903167534436508 T^{19} + 56161249804004502 T^{20} + 55573881576866490 T^{21} - 1071803450698157928 T^{22} - 4795411055555611122 T^{23} - 18444110399566611183 T^{24} - 97172652768258663054 T^{25} +$$$$47\!\cdots\!40$$$$T^{26} +$$$$86\!\cdots\!94$$$$T^{27} +$$$$25\!\cdots\!74$$$$T^{28} -$$$$15\!\cdots\!32$$$$T^{29} -$$$$10\!\cdots\!24$$$$T^{30} +$$$$66\!\cdots\!86$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$)($$1 - T - 188 T^{2} + 55 T^{3} + 18431 T^{4} + 6958 T^{5} - 1220598 T^{6} - 1466820 T^{7} + 61023285 T^{8} + 128763363 T^{9} - 2426865150 T^{10} - 7087005807 T^{11} + 80710069782 T^{12} + 247085609469 T^{13} - 2444398029834 T^{14} - 3801878390997 T^{15} + 81128049964254 T^{16} - 140669500466889 T^{17} - 3346380902842746 T^{18} + 12515627376433257 T^{19} + 151263665092702902 T^{20} - 491441025939358299 T^{21} - 6226672006436746350 T^{22} + 12223747755547878279 T^{23} +$$$$21\!\cdots\!85$$$$T^{24} -$$$$19\!\cdots\!40$$$$T^{25} -$$$$58\!\cdots\!02$$$$T^{26} +$$$$12\!\cdots\!54$$$$T^{27} +$$$$12\!\cdots\!11$$$$T^{28} +$$$$13\!\cdots\!35$$$$T^{29} -$$$$16\!\cdots\!32$$$$T^{30} -$$$$33\!\cdots\!93$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$)
$41$ ($$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 - 9 T - 34 T^{2} + 747 T^{3} - 2085 T^{4} - 20394 T^{5} + 110775 T^{6} + 623979 T^{7} - 5992218 T^{8} - 18494757 T^{9} + 381591615 T^{10} - 758285037 T^{11} - 10072918458 T^{12} + 43005256659 T^{13} + 313023674775 T^{14} - 2362771363194 T^{15} - 9903967342485 T^{16} + 145481442589107 T^{17} - 271487457790114 T^{18} - 2946437409545649 T^{19} + 13422659310152401 T^{20}$$)($$1 + 6 T - 223 T^{2} - 1686 T^{3} + 25980 T^{4} + 231654 T^{5} - 1971341 T^{6} - 20408106 T^{7} + 109216031 T^{8} + 1268388768 T^{9} - 4836103872 T^{10} - 57438167556 T^{11} + 192545389345 T^{12} + 1824668193534 T^{13} - 7677147470143 T^{14} - 27964729912410 T^{15} + 313424888729076 T^{16} - 1146553926408810 T^{17} - 12905284897310383 T^{18} + 125757956566556814 T^{19} + 544087251940916545 T^{20} - 6654567885439614756 T^{21} - 22971997512303721152 T^{22} +$$$$24\!\cdots\!08$$$$T^{23} +$$$$87\!\cdots\!51$$$$T^{24} -$$$$66\!\cdots\!66$$$$T^{25} -$$$$26\!\cdots\!41$$$$T^{26} +$$$$12\!\cdots\!14$$$$T^{27} +$$$$58\!\cdots\!80$$$$T^{28} -$$$$15\!\cdots\!06$$$$T^{29} -$$$$84\!\cdots\!03$$$$T^{30} +$$$$93\!\cdots\!06$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$)($$1 + 6 T - 142 T^{2} - 1146 T^{3} + 7575 T^{4} + 89148 T^{5} - 186515 T^{6} - 3858942 T^{7} + 5165546 T^{8} + 137269398 T^{9} - 383379603 T^{10} - 6275799867 T^{11} + 14048576428 T^{12} + 262470296220 T^{13} + 63020850350 T^{14} - 4818424458177 T^{15} - 18114488698896 T^{16} - 197555402785257 T^{17} + 105938049438350 T^{18} + 18089715285778620 T^{19} + 39697919375761708 T^{20} - 727090330826925267 T^{21} - 1821093078123196323 T^{22} + 26733801933571993638 T^{23} + 41246498577585065066 T^{24} -$$$$12\!\cdots\!62$$$$T^{25} -$$$$25\!\cdots\!15$$$$T^{26} +$$$$49\!\cdots\!68$$$$T^{27} +$$$$17\!\cdots\!75$$$$T^{28} -$$$$10\!\cdots\!66$$$$T^{29} -$$$$53\!\cdots\!62$$$$T^{30} +$$$$93\!\cdots\!06$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$)
$43$ ($$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$)($$1 + 3 T - 79 T^{2} - 1100 T^{3} + 1674 T^{4} + 79931 T^{5} + 324899 T^{6} - 3229902 T^{7} - 28512986 T^{8} + 52724394 T^{9} + 1438527201 T^{10} + 2267148942 T^{11} - 52720511114 T^{12} - 256799818314 T^{13} + 1110765026099 T^{14} + 11750531857433 T^{15} + 10581961744026 T^{16} - 299000472217700 T^{17} - 923367821930479 T^{18} + 1507777835810529 T^{19} + 21611482313284249 T^{20}$$)($$1 - 2 T - 209 T^{2} + 602 T^{3} + 20774 T^{4} - 72052 T^{5} - 1457073 T^{6} + 4933350 T^{7} + 91147851 T^{8} - 235722522 T^{9} - 5466637218 T^{10} + 9144255228 T^{11} + 301177025103 T^{12} - 290829011802 T^{13} - 14773623661707 T^{14} + 4808085837138 T^{15} + 658882369500660 T^{16} + 206747690996934 T^{17} - 27316430150496243 T^{18} - 23122942241341614 T^{19} + 1029664314599161503 T^{20} + 1344282723462890004 T^{21} - 34556598512153357682 T^{22} - 64073768536679251854 T^{23} +$$$$10\!\cdots\!51$$$$T^{24} +$$$$24\!\cdots\!50$$$$T^{25} -$$$$31\!\cdots\!77$$$$T^{26} -$$$$66\!\cdots\!64$$$$T^{27} +$$$$83\!\cdots\!74$$$$T^{28} +$$$$10\!\cdots\!86$$$$T^{29} -$$$$15\!\cdots\!41$$$$T^{30} -$$$$63\!\cdots\!14$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$)($$1 - 2 T - 137 T^{2} + 620 T^{3} + 8660 T^{4} - 62377 T^{5} - 131688 T^{6} + 3324402 T^{7} - 16902063 T^{8} - 47132751 T^{9} + 1398120027 T^{10} - 5063156292 T^{11} - 36164979717 T^{12} + 388120390392 T^{13} - 815732360646 T^{14} - 8046953157708 T^{15} + 93229603031718 T^{16} - 346018985781444 T^{17} - 1508289134834454 T^{18} + 30858287878896744 T^{19} - 123640868821459317 T^{20} - 744326723152573356 T^{21} + 8838024276744682323 T^{22} - 12811558914472065357 T^{23} -$$$$19\!\cdots\!63$$$$T^{24} +$$$$16\!\cdots\!86$$$$T^{25} -$$$$28\!\cdots\!12$$$$T^{26} -$$$$57\!\cdots\!39$$$$T^{27} +$$$$34\!\cdots\!60$$$$T^{28} +$$$$10\!\cdots\!60$$$$T^{29} -$$$$10\!\cdots\!13$$$$T^{30} -$$$$63\!\cdots\!14$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$)
$47$ ($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 15 T + 274 T^{2} + 2589 T^{3} + 26755 T^{4} + 176529 T^{5} + 1257485 T^{6} + 5719101 T^{7} + 28447502 T^{8} + 73195215 T^{9} + 229345007 T^{10} )^{2}$$)($$( 1 - 18 T + 379 T^{2} - 4272 T^{3} + 53788 T^{4} - 467070 T^{5} + 4528654 T^{6} - 32587152 T^{7} + 257763508 T^{8} - 1531596144 T^{9} + 10003796686 T^{10} - 48492608610 T^{11} + 262468281628 T^{12} - 979761869904 T^{13} + 4085322609691 T^{14} - 9119216168334 T^{15} + 23811286661761 T^{16} )^{2}$$)($$( 1 - 18 T + 262 T^{2} - 2454 T^{3} + 20893 T^{4} - 154293 T^{5} + 1267261 T^{6} - 9657933 T^{7} + 72865294 T^{8} - 453922851 T^{9} + 2799379549 T^{10} - 16019162139 T^{11} + 101951175133 T^{12} - 562812647178 T^{13} + 2824154416198 T^{14} - 9119216168334 T^{15} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$1 - 15 T + 128 T^{2} - 795 T^{3} + 2809 T^{4}$$)($$1 - 9 T + 237 T^{2} - 1890 T^{3} + 28720 T^{4} - 208689 T^{5} + 2523571 T^{6} - 16852668 T^{7} + 178203742 T^{8} - 1088604978 T^{9} + 10361882797 T^{10} - 57696063834 T^{11} + 500574311278 T^{12} - 2508974653836 T^{13} + 19912189027651 T^{14} - 87272799238677 T^{15} + 636560451624880 T^{16} - 2220204054291930 T^{17} + 14755546627492557 T^{18} - 29697872326219197 T^{19} + 174887470365513049 T^{20}$$)($$1 + 36 T + 766 T^{2} + 12024 T^{3} + 153999 T^{4} + 1662408 T^{5} + 15312632 T^{6} + 121481640 T^{7} + 835010600 T^{8} + 5038052724 T^{9} + 28358007246 T^{10} + 182105031078 T^{11} + 1622318702806 T^{12} + 17289317216124 T^{13} + 177226593270661 T^{14} + 1605696832118286 T^{15} + 12594715934262750 T^{16} + 85101932102269158 T^{17} + 497829500497286749 T^{18} + 2573981679184892748 T^{19} + 12800874900435389686 T^{20} + 76155503249444531454 T^{21} +$$$$62\!\cdots\!34$$$$T^{22} +$$$$59\!\cdots\!88$$$$T^{23} +$$$$51\!\cdots\!00$$$$T^{24} +$$$$40\!\cdots\!20$$$$T^{25} +$$$$26\!\cdots\!68$$$$T^{26} +$$$$15\!\cdots\!76$$$$T^{27} +$$$$75\!\cdots\!59$$$$T^{28} +$$$$31\!\cdots\!52$$$$T^{29} +$$$$10\!\cdots\!54$$$$T^{30} +$$$$26\!\cdots\!52$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$)($$1 + 271 T^{2} + 39798 T^{4} + 8019 T^{5} + 3882470 T^{6} + 3112830 T^{7} + 272618249 T^{8} + 573793713 T^{9} + 13751201223 T^{10} + 69938411256 T^{11} + 461723008303 T^{12} + 6218775316710 T^{13} + 7202514917530 T^{14} + 423085276358208 T^{15} - 24546838659498 T^{16} + 22423519646985024 T^{17} + 20231864403341770 T^{18} + 925832612825834670 T^{19} + 3643216624277663743 T^{20} + 29247928374839669208 T^{21} +$$$$30\!\cdots\!67$$$$T^{22} +$$$$67\!\cdots\!81$$$$T^{23} +$$$$16\!\cdots\!89$$$$T^{24} +$$$$10\!\cdots\!90$$$$T^{25} +$$$$67\!\cdots\!30$$$$T^{26} +$$$$74\!\cdots\!43$$$$T^{27} +$$$$19\!\cdots\!18$$$$T^{28} +$$$$37\!\cdots\!99$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$)
$59$ ($$( 1 + 59 T^{2} )^{2}$$)($$( 1 - 18 T + 379 T^{2} - 4179 T^{3} + 48448 T^{4} - 365781 T^{5} + 2858432 T^{6} - 14547099 T^{7} + 77838641 T^{8} - 218112498 T^{9} + 714924299 T^{10} )^{2}$$)($$( 1 + 30 T + 700 T^{2} + 11136 T^{3} + 154015 T^{4} + 1732056 T^{5} + 17715805 T^{6} + 156334302 T^{7} + 1281973738 T^{8} + 9223723818 T^{9} + 61668717205 T^{10} + 355727929224 T^{11} + 1866255354415 T^{12} + 7961396993664 T^{13} + 29526373548700 T^{14} + 74659544544570 T^{15} + 146830437604321 T^{16} )^{2}$$)($$( 1 - 15 T + 304 T^{2} - 2994 T^{3} + 35017 T^{4} - 245181 T^{5} + 2158909 T^{6} - 12195126 T^{7} + 111191710 T^{8} - 719512434 T^{9} + 7515162229 T^{10} - 50355028599 T^{11} + 424313630137 T^{12} - 2140483351206 T^{13} + 12822882226864 T^{14} - 37329772272285 T^{15} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$1 + 70 T^{2} + 3721 T^{4}$$)($$1 - 358 T^{2} + 54069 T^{4} - 4372581 T^{6} + 210774216 T^{8} - 9493721499 T^{10} + 784290857736 T^{12} - 60542061285621 T^{14} + 2785655121324909 T^{16} - 68631218053026598 T^{18} + 713342911662882601 T^{20}$$)($$1 - 472 T^{2} + 107634 T^{4} - 15742946 T^{6} + 1661711189 T^{8} - 136358974992 T^{10} + 9312516091879 T^{12} - 573351459745642 T^{14} + 34616863644604218 T^{16} - 2133440781713533882 T^{18} +$$$$12\!\cdots\!39$$$$T^{20} -$$$$70\!\cdots\!12$$$$T^{22} +$$$$31\!\cdots\!09$$$$T^{24} -$$$$11\!\cdots\!46$$$$T^{26} +$$$$28\!\cdots\!14$$$$T^{28} -$$$$46\!\cdots\!52$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$)($$1 - 745 T^{2} + 270666 T^{4} - 63553928 T^{6} + 10780728317 T^{8} - 1399459132305 T^{10} + 143698232661445 T^{12} - 11899376361024010 T^{14} + 802850906302349022 T^{16} - 44277579439370341210 T^{18} +$$$$19\!\cdots\!45$$$$T^{20} -$$$$72\!\cdots\!05$$$$T^{22} +$$$$20\!\cdots\!77$$$$T^{24} -$$$$45\!\cdots\!28$$$$T^{26} +$$$$71\!\cdots\!86$$$$T^{28} -$$$$73\!\cdots\!45$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$)
$67$ ($$( 1 - 4 T + 67 T^{2} )^{2}$$)($$( 1 + 10 T + 282 T^{2} + 2634 T^{3} + 34317 T^{4} + 263157 T^{5} + 2299239 T^{6} + 11824026 T^{7} + 84815166 T^{8} + 201511210 T^{9} + 1350125107 T^{10} )^{2}$$)($$( 1 - 14 T + 435 T^{2} - 5596 T^{3} + 90428 T^{4} - 1008078 T^{5} + 11375926 T^{6} - 106594220 T^{7} + 933924096 T^{8} - 7141812740 T^{9} + 51066531814 T^{10} - 303192563514 T^{11} + 1822225569788 T^{12} - 7555300098772 T^{13} + 39349396243515 T^{14} - 84849962474522 T^{15} + 406067677556641 T^{16} )^{2}$$)($$( 1 + 7 T + 315 T^{2} + 1580 T^{3} + 47600 T^{4} + 182439 T^{5} + 4735936 T^{6} + 14472934 T^{7} + 354986784 T^{8} + 969686578 T^{9} + 21259616704 T^{10} + 54870900957 T^{11} + 959193359600 T^{12} + 2133197669060 T^{13} + 28494390383235 T^{14} + 42424981237261 T^{15} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$1 - 130 T^{2} + 5041 T^{4}$$)($$1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 3769705738633 T^{12} - 198234871721735 T^{14} + 8075057597528077 T^{16} - 226659489467262111 T^{18} + 3255243551009881201 T^{20}$$)($$1 - 650 T^{2} + 199389 T^{4} - 38930632 T^{6} + 5566228364 T^{8} - 640511863116 T^{10} + 63163988645884 T^{12} - 5475157404521894 T^{14} + 416179213677825948 T^{16} - 27600268476194867654 T^{18} +$$$$16\!\cdots\!04$$$$T^{20} -$$$$82\!\cdots\!36$$$$T^{22} +$$$$35\!\cdots\!04$$$$T^{24} -$$$$12\!\cdots\!32$$$$T^{26} +$$$$32\!\cdots\!49$$$$T^{28} -$$$$53\!\cdots\!50$$$$T^{30} +$$$$41\!\cdots\!21$$$$T^{32}$$)($$1 - 389 T^{2} + 95718 T^{4} - 16982206 T^{6} + 2417461733 T^{8} - 285171142929 T^{10} + 28739565041956 T^{12} - 2501948752776044 T^{14} + 190114711967546784 T^{16} - 12612323662744037804 T^{18} +$$$$73\!\cdots\!36$$$$T^{20} -$$$$36\!\cdots\!09$$$$T^{22} +$$$$15\!\cdots\!13$$$$T^{24} -$$$$55\!\cdots\!06$$$$T^{26} +$$$$15\!\cdots\!38$$$$T^{28} -$$$$32\!\cdots\!09$$$$T^{30} +$$$$41\!\cdots\!21$$$$T^{32}$$)
$73$ ($$1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4}$$)($$1 - 3 T + 260 T^{2} - 771 T^{3} + 36639 T^{4} - 155724 T^{5} + 3663555 T^{6} - 21043473 T^{7} + 293486934 T^{8} - 2074196103 T^{9} + 21799556757 T^{10} - 151416315519 T^{11} + 1563991871286 T^{12} - 8186268736041 T^{13} + 104038517806755 T^{14} - 322827000748332 T^{15} + 5544734717002671 T^{16} - 8517544258223787 T^{17} + 209679623892461060 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20}$$)($$1 + 434 T^{2} + 101253 T^{4} + 70380 T^{5} + 16361620 T^{6} + 29833866 T^{7} + 2027081504 T^{8} + 6817588704 T^{9} + 204710173188 T^{10} + 1064159524062 T^{11} + 17706469884892 T^{12} + 124668495857988 T^{13} + 1383495353685575 T^{14} + 11427374402737884 T^{15} + 102528286806790386 T^{16} + 834198331399865532 T^{17} + 7372646739790429175 T^{18} + 48498164253186917796 T^{19} +$$$$50\!\cdots\!72$$$$T^{20} +$$$$22\!\cdots\!66$$$$T^{21} +$$$$30\!\cdots\!32$$$$T^{22} +$$$$75\!\cdots\!88$$$$T^{23} +$$$$16\!\cdots\!24$$$$T^{24} +$$$$17\!\cdots\!58$$$$T^{25} +$$$$70\!\cdots\!80$$$$T^{26} +$$$$22\!\cdots\!60$$$$T^{27} +$$$$23\!\cdots\!13$$$$T^{28} +$$$$52\!\cdots\!06$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$)($$1 + 278 T^{2} + 39699 T^{4} - 160362 T^{5} + 4049527 T^{6} - 43185312 T^{7} + 316493258 T^{8} - 5978310966 T^{9} + 30516141699 T^{10} - 573490190745 T^{11} + 3892519182646 T^{12} - 40210747978752 T^{13} + 436575836052170 T^{14} - 2488085380264887 T^{15} + 37348173312201540 T^{16} - 181630232759336751 T^{17} + 2326512630322013930 T^{18} - 15642664546450166784 T^{19} +$$$$11\!\cdots\!86$$$$T^{20} -$$$$11\!\cdots\!85$$$$T^{21} +$$$$46\!\cdots\!11$$$$T^{22} -$$$$66\!\cdots\!02$$$$T^{23} +$$$$25\!\cdots\!98$$$$T^{24} -$$$$25\!\cdots\!56$$$$T^{25} +$$$$17\!\cdots\!23$$$$T^{26} -$$$$50\!\cdots\!74$$$$T^{27} +$$$$90\!\cdots\!79$$$$T^{28} +$$$$33\!\cdots\!02$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$)
$79$ ($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 20 T + 354 T^{2} - 3468 T^{3} + 34803 T^{4} - 265629 T^{5} + 2749437 T^{6} - 21643788 T^{7} + 174535806 T^{8} - 779001620 T^{9} + 3077056399 T^{10} )^{2}$$)($$( 1 + 16 T + 483 T^{2} + 6542 T^{3} + 113126 T^{4} + 1278384 T^{5} + 16380706 T^{6} + 152801626 T^{7} + 1573503228 T^{8} + 12071328454 T^{9} + 102231986146 T^{10} + 630293168976 T^{11} + 4406266863206 T^{12} + 20130102962258 T^{13} + 117411241016643 T^{14} + 307262543778544 T^{15} + 1517108809906561 T^{16} )^{2}$$)($$( 1 + T + 489 T^{2} + 92 T^{3} + 108962 T^{4} - 44511 T^{5} + 14856838 T^{6} - 9973310 T^{7} + 1391605452 T^{8} - 787891490 T^{9} + 92721525958 T^{10} - 21945658929 T^{11} + 4244078725922 T^{12} + 283089188708 T^{13} + 118869765749769 T^{14} + 19203908986159 T^{15} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4}$$)($$1 + 15 T - 136 T^{2} - 1773 T^{3} + 22674 T^{4} + 93717 T^{5} - 3687774 T^{6} - 10067337 T^{7} + 346135869 T^{8} + 496605294 T^{9} - 27460905396 T^{10} + 41218239402 T^{11} + 2384530001541 T^{12} - 5756372421219 T^{13} - 175015562267454 T^{14} + 369155071940031 T^{15} + 7413046025768706 T^{16} - 48112218404608671 T^{17} - 306311743570909576 T^{18} + 2804103829013106045 T^{19} + 15516041187205853449 T^{20}$$)($$1 - 487 T^{2} - 312 T^{3} + 123774 T^{4} + 132990 T^{5} - 22183883 T^{6} - 29138634 T^{7} + 3170469341 T^{8} + 4110229572 T^{9} - 386467088226 T^{10} - 393115428402 T^{11} + 41656592194789 T^{12} + 25282823866380 T^{13} - 4030130568645907 T^{14} - 780079655467782 T^{15} + 351851707607703156 T^{16} - 64746611403825906 T^{17} - 27763569487401653323 T^{18} + 14456390010085821060 T^{19} +$$$$19\!\cdots\!69$$$$T^{20} -$$$$15\!\cdots\!86$$$$T^{21} -$$$$12\!\cdots\!94$$$$T^{22} +$$$$11\!\cdots\!44$$$$T^{23} +$$$$71\!\cdots\!81$$$$T^{24} -$$$$54\!\cdots\!02$$$$T^{25} -$$$$34\!\cdots\!67$$$$T^{26} +$$$$17\!\cdots\!30$$$$T^{27} +$$$$13\!\cdots\!14$$$$T^{28} -$$$$27\!\cdots\!56$$$$T^{29} -$$$$35\!\cdots\!23$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$)($$1 - 334 T^{2} + 1272 T^{3} + 54816 T^{4} - 395796 T^{5} - 5397056 T^{6} + 60133572 T^{7} + 310945691 T^{8} - 6455120304 T^{9} - 3389068248 T^{10} + 563598757704 T^{11} - 2368219763012 T^{12} - 39202056035964 T^{13} + 437295746267318 T^{14} + 1352773893773628 T^{15} - 45259898774669136 T^{16} + 112280233183211124 T^{17} + 3012530396035553702 T^{18} - 22415226014635747668 T^{19} -$$$$11\!\cdots\!52$$$$T^{20} +$$$$22\!\cdots\!72$$$$T^{21} -$$$$11\!\cdots\!12$$$$T^{22} -$$$$17\!\cdots\!08$$$$T^{23} +$$$$70\!\cdots\!31$$$$T^{24} +$$$$11\!\cdots\!16$$$$T^{25} -$$$$83\!\cdots\!44$$$$T^{26} -$$$$50\!\cdots\!32$$$$T^{27} +$$$$58\!\cdots\!76$$$$T^{28} +$$$$11\!\cdots\!36$$$$T^{29} -$$$$24\!\cdots\!86$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4}$$)($$1 + 24 T + 44 T^{2} - 2592 T^{3} - 1287 T^{4} + 278721 T^{5} - 235110 T^{6} - 13705920 T^{7} + 186157425 T^{8} + 904992183 T^{9} - 14602879521 T^{10} + 80544304287 T^{11} + 1474552963425 T^{12} - 9662248716480 T^{13} - 14751328281510 T^{14} + 1556394633684729 T^{15} - 639614921466807 T^{16} - 114647620049211168 T^{17} + 173209907450891564 T^{18} + 8408553688979645016 T^{19} + 31181719929966183601 T^{20}$$)($$1 + 24 T - 10 T^{2} - 3636 T^{3} + 1197 T^{4} + 543420 T^{5} + 1792912 T^{6} - 58468350 T^{7} - 544062388 T^{8} + 3510821196 T^{9} + 61278252588 T^{10} - 296095488138 T^{11} - 6613804791944 T^{12} + 16297764313308 T^{13} + 597103648531739 T^{14} + 218011613471172 T^{15} - 39896652829458150 T^{16} + 19403033598934308 T^{17} + 4729658000019904619 T^{18} + 11489418610188427452 T^{19} -$$$$41\!\cdots\!04$$$$T^{20} -$$$$16\!\cdots\!62$$$$T^{21} +$$$$30\!\cdots\!68$$$$T^{22} +$$$$15\!\cdots\!84$$$$T^{23} -$$$$21\!\cdots\!28$$$$T^{24} -$$$$20\!\cdots\!50$$$$T^{25} +$$$$55\!\cdots\!12$$$$T^{26} +$$$$15\!\cdots\!80$$$$T^{27} +$$$$29\!\cdots\!37$$$$T^{28} -$$$$79\!\cdots\!84$$$$T^{29} -$$$$19\!\cdots\!10$$$$T^{30} +$$$$41\!\cdots\!76$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$)($$1 - 21 T - 253 T^{2} + 6084 T^{3} + 64134 T^{4} - 1086543 T^{5} - 14444834 T^{6} + 154075794 T^{7} + 2457441293 T^{8} - 16916386392 T^{9} - 343956627405 T^{10} + 1437006555873 T^{11} + 41653185355711 T^{12} - 92739461592729 T^{13} - 4399434577615834 T^{14} + 3097874885577318 T^{15} + 411982470004053510 T^{16} + 275710864816381302 T^{17} - 34847921289295021114 T^{18} - 65378445499564570401 T^{19} +$$$$26\!\cdots\!51$$$$T^{20} +$$$$80\!\cdots\!77$$$$T^{21} -$$$$17\!\cdots\!05$$$$T^{22} -$$$$74\!\cdots\!68$$$$T^{23} +$$$$96\!\cdots\!33$$$$T^{24} +$$$$53\!\cdots\!46$$$$T^{25} -$$$$45\!\cdots\!34$$$$T^{26} -$$$$30\!\cdots\!27$$$$T^{27} +$$$$15\!\cdots\!14$$$$T^{28} +$$$$13\!\cdots\!96$$$$T^{29} -$$$$49\!\cdots\!73$$$$T^{30} -$$$$36\!\cdots\!29$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$)
$97$ ($$1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4}$$)($$1 + 6 T + 311 T^{2} + 1794 T^{3} + 51903 T^{4} + 385032 T^{5} + 6010353 T^{6} + 63309837 T^{7} + 574248354 T^{8} + 8264282925 T^{9} + 54719955099 T^{10} + 801635443725 T^{11} + 5403102762786 T^{12} + 57781178864301 T^{13} + 532092229646193 T^{14} + 3306400793833224 T^{15} + 43233745971829887 T^{16} + 144952122353734722 T^{17} + 2437441847851234871 T^{18} + 4561386351927391302 T^{19} + 73742412689492826049 T^{20}$$)($$1 - 6 T + 395 T^{2} - 2298 T^{3} + 68379 T^{4} - 571872 T^{5} + 8922340 T^{6} - 109039644 T^{7} + 1233764039 T^{8} - 14418445794 T^{9} + 171203907675 T^{10} - 1605167799666 T^{11} + 21002714173786 T^{12} - 194363495836338 T^{13} + 2133943680044999 T^{14} - 22669145285986746 T^{15} + 200280216945639852 T^{16} - 2198907092740714362 T^{17} + 20078276085543395591 T^{18} -$$$$17\!\cdots\!74$$$$T^{19} +$$$$18\!\cdots\!66$$$$T^{20} -$$$$13\!\cdots\!62$$$$T^{21} +$$$$14\!\cdots\!75$$$$T^{22} -$$$$11\!\cdots\!22$$$$T^{23} +$$$$96\!\cdots\!79$$$$T^{24} -$$$$82\!\cdots\!48$$$$T^{25} +$$$$65\!\cdots\!60$$$$T^{26} -$$$$40\!\cdots\!16$$$$T^{27} +$$$$47\!\cdots\!39$$$$T^{28} -$$$$15\!\cdots\!46$$$$T^{29} +$$$$25\!\cdots\!55$$$$T^{30} -$$$$37\!\cdots\!58$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$)($$1 + 3 T + 392 T^{2} + 1167 T^{3} + 75243 T^{4} + 248802 T^{5} + 9281977 T^{6} + 31870980 T^{7} + 787182476 T^{8} + 2086019346 T^{9} + 40966656945 T^{10} - 99548858823 T^{11} - 1131878124686 T^{12} - 48785193558327 T^{13} - 618766130152816 T^{14} - 7413615555033420 T^{15} - 81803373889459032 T^{16} - 719120708838241740 T^{17} - 5821970518607845744 T^{18} - 44524928960458978071 T^{19} -$$$$10\!\cdots\!66$$$$T^{20} -$$$$85\!\cdots\!11$$$$T^{21} +$$$$34\!\cdots\!05$$$$T^{22} +$$$$16\!\cdots\!98$$$$T^{23} +$$$$61\!\cdots\!36$$$$T^{24} +$$$$24\!\cdots\!60$$$$T^{25} +$$$$68\!\cdots\!73$$$$T^{26} +$$$$17\!\cdots\!06$$$$T^{27} +$$$$52\!\cdots\!63$$$$T^{28} +$$$$78\!\cdots\!59$$$$T^{29} +$$$$25\!\cdots\!48$$$$T^{30} +$$$$18\!\cdots\!79$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$)