# Properties

 Label 671.2.a Level $671$ Weight $2$ Character orbit 671.a Rep. character $\chi_{671}(1,\cdot)$ Character field $\Q$ Dimension $51$ Newform subspaces $4$ Sturm bound $124$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$671 = 11 \cdot 61$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 671.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$124$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(671))$$.

Total New Old
Modular forms 64 51 13
Cusp forms 61 51 10
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$11$$$$61$$FrickeDim.
$$+$$$$+$$$$+$$$$6$$
$$+$$$$-$$$$-$$$$21$$
$$-$$$$+$$$$-$$$$19$$
$$-$$$$-$$$$+$$$$5$$
Plus space$$+$$$$11$$
Minus space$$-$$$$40$$

## Trace form

 $$51q + 3q^{2} + 2q^{3} + 57q^{4} + 4q^{5} + 8q^{7} - 9q^{8} + 61q^{9} + O(q^{10})$$ $$51q + 3q^{2} + 2q^{3} + 57q^{4} + 4q^{5} + 8q^{7} - 9q^{8} + 61q^{9} + 6q^{10} - 3q^{11} + 8q^{12} + 14q^{13} + 4q^{14} + 2q^{15} + 73q^{16} + 2q^{17} - q^{18} + 16q^{19} - 14q^{20} + 20q^{21} + 3q^{22} - 2q^{23} + 8q^{24} + 67q^{25} + 10q^{26} - 34q^{27} + 12q^{28} + 10q^{29} - 12q^{30} + 6q^{31} + 15q^{32} - 2q^{33} + 6q^{34} - 28q^{35} + 101q^{36} + 16q^{37} - 20q^{38} - 8q^{39} - 6q^{40} + 10q^{41} - 20q^{42} + 36q^{43} - 11q^{44} + 22q^{45} + 24q^{46} - 12q^{47} - 64q^{48} + 91q^{49} + 29q^{50} + 32q^{51} + 22q^{52} + 6q^{53} - 16q^{54} - 8q^{55} + 24q^{56} + 44q^{57} - 14q^{58} - 34q^{59} - 28q^{60} + q^{61} + 4q^{63} + 113q^{64} + 48q^{65} - 8q^{66} + 22q^{67} - 18q^{68} - 34q^{69} - 28q^{70} - 18q^{71} - 37q^{72} + 22q^{73} - 26q^{74} - 52q^{75} + 4q^{76} + 8q^{77} - 112q^{78} + 48q^{79} - 50q^{80} + 123q^{81} - 42q^{82} - 4q^{83} - 60q^{84} + 56q^{85} - 32q^{86} - 36q^{87} + 15q^{88} - 126q^{90} + 48q^{91} + 12q^{92} - 42q^{93} + 12q^{94} - 36q^{95} - 44q^{96} + 60q^{97} - 77q^{98} - 9q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(671))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 11 61
671.2.a.a $$5$$ $$5.358$$ 5.5.24217.1 None $$-2$$ $$0$$ $$-2$$ $$-1$$ $$-$$ $$-$$ $$q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{4})q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots$$
671.2.a.b $$6$$ $$5.358$$ 6.6.2661761.1 None $$0$$ $$-1$$ $$-1$$ $$-5$$ $$+$$ $$+$$ $$q+\beta _{3}q^{2}-\beta _{1}q^{3}-\beta _{5}q^{4}+(\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots$$
671.2.a.c $$19$$ $$5.358$$ $$\mathbb{Q}[x]/(x^{19} - \cdots)$$ None $$5$$ $$0$$ $$0$$ $$9$$ $$-$$ $$+$$ $$q+\beta _{1}q^{2}+\beta _{11}q^{3}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+\cdots$$
671.2.a.d $$21$$ $$5.358$$ None $$0$$ $$3$$ $$7$$ $$5$$ $$+$$ $$-$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(671))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(671)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(61))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T + 7 T^{2} + 12 T^{3} + 24 T^{4} + 33 T^{5} + 48 T^{6} + 48 T^{7} + 56 T^{8} + 32 T^{9} + 32 T^{10}$$)($$1 + 5 T^{2} + 2 T^{3} + 16 T^{4} + 7 T^{5} + 38 T^{6} + 14 T^{7} + 64 T^{8} + 16 T^{9} + 80 T^{10} + 64 T^{12}$$)($$1 - 5 T + 20 T^{2} - 58 T^{3} + 150 T^{4} - 333 T^{5} + 687 T^{6} - 1281 T^{7} + 2245 T^{8} - 3601 T^{9} + 5389 T^{10} - 7263 T^{11} + 8807 T^{12} - 8713 T^{13} + 5731 T^{14} + 2894 T^{15} - 18702 T^{16} + 44917 T^{17} - 79669 T^{18} + 122313 T^{19} - 159338 T^{20} + 179668 T^{21} - 149616 T^{22} + 46304 T^{23} + 183392 T^{24} - 557632 T^{25} + 1127296 T^{26} - 1859328 T^{27} + 2759168 T^{28} - 3687424 T^{29} + 4597760 T^{30} - 5246976 T^{31} + 5627904 T^{32} - 5455872 T^{33} + 4915200 T^{34} - 3801088 T^{35} + 2621440 T^{36} - 1310720 T^{37} + 524288 T^{38}$$)
$3$ ($$1 + 9 T^{2} + 3 T^{3} + 38 T^{4} + 17 T^{5} + 114 T^{6} + 27 T^{7} + 243 T^{8} + 243 T^{10}$$)($$1 + T + 12 T^{2} + 12 T^{3} + 72 T^{4} + 64 T^{5} + 269 T^{6} + 192 T^{7} + 648 T^{8} + 324 T^{9} + 972 T^{10} + 243 T^{11} + 729 T^{12}$$)($$1 + 14 T^{2} + 11 T^{3} + 108 T^{4} + 160 T^{5} + 686 T^{6} + 1231 T^{7} + 3921 T^{8} + 7063 T^{9} + 19708 T^{10} + 34531 T^{11} + 86296 T^{12} + 149743 T^{13} + 337316 T^{14} + 575599 T^{15} + 1208364 T^{16} + 1976651 T^{17} + 3974264 T^{18} + 6184547 T^{19} + 11922792 T^{20} + 17789859 T^{21} + 32625828 T^{22} + 46623519 T^{23} + 81967788 T^{24} + 109162647 T^{25} + 188729352 T^{26} + 226557891 T^{27} + 387912564 T^{28} + 417063087 T^{29} + 694593387 T^{30} + 654203871 T^{31} + 1093705578 T^{32} + 765275040 T^{33} + 1549681956 T^{34} + 473513931 T^{35} + 1807962282 T^{36} + 1162261467 T^{38}$$)
$5$ ($$1 + 2 T + 22 T^{2} + 34 T^{3} + 205 T^{4} + 241 T^{5} + 1025 T^{6} + 850 T^{7} + 2750 T^{8} + 1250 T^{9} + 3125 T^{10}$$)($$1 + T + 21 T^{2} + 22 T^{3} + 201 T^{4} + 206 T^{5} + 1209 T^{6} + 1030 T^{7} + 5025 T^{8} + 2750 T^{9} + 13125 T^{10} + 3125 T^{11} + 15625 T^{12}$$)($$1 + 25 T^{2} + 20 T^{3} + 364 T^{4} + 535 T^{5} + 4067 T^{6} + 7933 T^{7} + 38993 T^{8} + 85681 T^{9} + 330223 T^{10} + 753981 T^{11} + 2490057 T^{12} + 5697247 T^{13} + 16785768 T^{14} + 37904374 T^{15} + 101886947 T^{16} + 224216574 T^{17} + 560432351 T^{18} + 1185201854 T^{19} + 2802161755 T^{20} + 5605414350 T^{21} + 12735868375 T^{22} + 23690233750 T^{23} + 52455525000 T^{24} + 89019484375 T^{25} + 194535703125 T^{26} + 294523828125 T^{27} + 644966796875 T^{28} + 836728515625 T^{29} + 1903955078125 T^{30} + 1936767578125 T^{31} + 4964599609375 T^{32} + 3265380859375 T^{33} + 11108398437500 T^{34} + 3051757812500 T^{35} + 19073486328125 T^{36} + 19073486328125 T^{38}$$)
$7$ ($$1 + T + 21 T^{2} + 27 T^{3} + 233 T^{4} + 257 T^{5} + 1631 T^{6} + 1323 T^{7} + 7203 T^{8} + 2401 T^{9} + 16807 T^{10}$$)($$1 + 5 T + 46 T^{2} + 164 T^{3} + 834 T^{4} + 2222 T^{5} + 7858 T^{6} + 15554 T^{7} + 40866 T^{8} + 56252 T^{9} + 110446 T^{10} + 84035 T^{11} + 117649 T^{12}$$)($$1 - 9 T + 86 T^{2} - 532 T^{3} + 3193 T^{4} - 15715 T^{5} + 74445 T^{6} - 312953 T^{7} + 1270779 T^{8} - 4747891 T^{9} + 17215303 T^{10} - 58598584 T^{11} + 194409144 T^{12} - 613151707 T^{13} + 1892262155 T^{14} - 5598730940 T^{15} + 16264737590 T^{16} - 45556518226 T^{17} + 125587504358 T^{18} - 334782347206 T^{19} + 879112530506 T^{20} - 2232269393074 T^{21} + 5578804993370 T^{22} - 13442552986940 T^{23} + 31803250039085 T^{24} - 72136685176843 T^{25} + 160104289677192 T^{26} - 337809175641784 T^{27} + 694699571647921 T^{28} - 1341161692449859 T^{29} + 2512745301142797 T^{30} - 4331672353414553 T^{31} + 7212902379749115 T^{32} - 10658275589822035 T^{33} + 15158963901247999 T^{34} - 17679919063027732 T^{35} + 20006224202899802 T^{36} - 14655722381194041 T^{37} + 11398895185373143 T^{38}$$)
$11$ ($$( 1 - T )^{5}$$)($$( 1 + T )^{6}$$)($$( 1 - T )^{19}$$)
$13$ ($$1 + 10 T + 99 T^{2} + 561 T^{3} + 3014 T^{4} + 11183 T^{5} + 39182 T^{6} + 94809 T^{7} + 217503 T^{8} + 285610 T^{9} + 371293 T^{10}$$)($$1 + 4 T + 66 T^{2} + 197 T^{3} + 1875 T^{4} + 4358 T^{5} + 30838 T^{6} + 56654 T^{7} + 316875 T^{8} + 432809 T^{9} + 1885026 T^{10} + 1485172 T^{11} + 4826809 T^{12}$$)($$1 - 8 T + 125 T^{2} - 879 T^{3} + 8055 T^{4} - 49937 T^{5} + 352375 T^{6} - 1957698 T^{7} + 11695722 T^{8} - 59256283 T^{9} + 313383086 T^{10} - 1465953928 T^{11} + 7038581274 T^{12} - 30646578609 T^{13} + 135623829486 T^{14} - 552635331582 T^{15} + 2275267941252 T^{16} - 8702950115787 T^{17} + 33531541709632 T^{18} - 120515386387602 T^{19} + 435910042225216 T^{20} - 1470798569568003 T^{21} + 4998763666930644 T^{22} - 15783817705313502 T^{23} + 50356178521345398 T^{24} - 147925181449128681 T^{25} + 441660536727470658 T^{26} - 1195823654640222088 T^{27} + 3323270738995805078 T^{28} - 8168981806957537267 T^{29} + 20960609748067209714 T^{30} - 45610614648110808738 T^{31} +$$$$10\!\cdots\!75$$$$T^{32} -$$$$19\!\cdots\!93$$$$T^{33} +$$$$41\!\cdots\!35$$$$T^{34} -$$$$58\!\cdots\!39$$$$T^{35} +$$$$10\!\cdots\!25$$$$T^{36} -$$$$89\!\cdots\!32$$$$T^{37} +$$$$14\!\cdots\!77$$$$T^{38}$$)
$17$ ($$1 + 3 T + 84 T^{2} + 199 T^{3} + 2839 T^{4} + 5033 T^{5} + 48263 T^{6} + 57511 T^{7} + 412692 T^{8} + 250563 T^{9} + 1419857 T^{10}$$)($$1 + 5 T + 63 T^{2} + 282 T^{3} + 2011 T^{4} + 7720 T^{5} + 40860 T^{6} + 131240 T^{7} + 581179 T^{8} + 1385466 T^{9} + 5261823 T^{10} + 7099285 T^{11} + 24137569 T^{12}$$)($$1 - 9 T + 155 T^{2} - 1150 T^{3} + 12186 T^{4} - 80063 T^{5} + 660999 T^{6} - 3950590 T^{7} + 27673665 T^{8} - 152720701 T^{9} + 947579608 T^{10} - 4871861803 T^{11} + 27433956192 T^{12} - 132108866795 T^{13} + 685419029810 T^{14} - 3101461999645 T^{15} + 14969290650336 T^{16} - 63739222793853 T^{17} + 287928271918558 T^{18} - 1153564810721111 T^{19} + 4894780622615486 T^{20} - 18420635387423517 T^{21} + 73544124965100768 T^{22} - 259037207672350045 T^{23} + 973197007408937170 T^{24} - 3188786887776121355 T^{25} + 11257213178965413216 T^{26} - 33984926223800926123 T^{27} +$$$$11\!\cdots\!76$$$$T^{28} -$$$$30\!\cdots\!49$$$$T^{29} +$$$$94\!\cdots\!45$$$$T^{30} -$$$$23\!\cdots\!90$$$$T^{31} +$$$$65\!\cdots\!63$$$$T^{32} -$$$$13\!\cdots\!27$$$$T^{33} +$$$$34\!\cdots\!98$$$$T^{34} -$$$$55\!\cdots\!50$$$$T^{35} +$$$$12\!\cdots\!35$$$$T^{36} -$$$$12\!\cdots\!81$$$$T^{37} +$$$$23\!\cdots\!53$$$$T^{38}$$)
$19$ ($$1 + 13 T + 144 T^{2} + 1024 T^{3} + 6356 T^{4} + 29521 T^{5} + 120764 T^{6} + 369664 T^{7} + 987696 T^{8} + 1694173 T^{9} + 2476099 T^{10}$$)($$1 + 3 T + 105 T^{2} + 245 T^{3} + 4688 T^{4} + 8533 T^{5} + 116050 T^{6} + 162127 T^{7} + 1692368 T^{8} + 1680455 T^{9} + 13683705 T^{10} + 7428297 T^{11} + 47045881 T^{12}$$)($$1 - 17 T + 288 T^{2} - 3268 T^{3} + 35007 T^{4} - 308746 T^{5} + 2573440 T^{6} - 18867802 T^{7} + 131543937 T^{8} - 831239199 T^{9} + 5027031863 T^{10} - 28015530644 T^{11} + 150536367591 T^{12} - 754277236101 T^{13} + 3683036143953 T^{14} - 16981481460006 T^{15} + 77564728648267 T^{16} - 339994280939897 T^{17} + 1506214010137031 T^{18} - 6497393486738832 T^{19} + 28618066192603589 T^{20} - 122737935419302817 T^{21} + 532016473798463353 T^{22} - 2213043645349441926 T^{23} + 9119562113005879347 T^{24} - 35485637090616549981 T^{25} +$$$$13\!\cdots\!49$$$$T^{26} -$$$$47\!\cdots\!04$$$$T^{27} +$$$$16\!\cdots\!77$$$$T^{28} -$$$$50\!\cdots\!99$$$$T^{29} +$$$$15\!\cdots\!03$$$$T^{30} -$$$$41\!\cdots\!22$$$$T^{31} +$$$$10\!\cdots\!60$$$$T^{32} -$$$$24\!\cdots\!66$$$$T^{33} +$$$$53\!\cdots\!93$$$$T^{34} -$$$$94\!\cdots\!08$$$$T^{35} +$$$$15\!\cdots\!32$$$$T^{36} -$$$$17\!\cdots\!97$$$$T^{37} +$$$$19\!\cdots\!79$$$$T^{38}$$)
$23$ ($$1 + 80 T^{2} + 101 T^{3} + 2826 T^{4} + 4587 T^{5} + 64998 T^{6} + 53429 T^{7} + 973360 T^{8} + 6436343 T^{10}$$)($$1 + 3 T + 95 T^{2} + 271 T^{3} + 4115 T^{4} + 11062 T^{5} + 113245 T^{6} + 254426 T^{7} + 2176835 T^{8} + 3297257 T^{9} + 26584895 T^{10} + 19309029 T^{11} + 148035889 T^{12}$$)($$1 + 10 T + 230 T^{2} + 2089 T^{3} + 27593 T^{4} + 227939 T^{5} + 2262592 T^{6} + 17086886 T^{7} + 141160625 T^{8} + 979720939 T^{9} + 7085381547 T^{10} + 45421660648 T^{11} + 295739078543 T^{12} + 1758638785657 T^{13} + 10479904969397 T^{14} + 58000348761914 T^{15} + 319464719445611 T^{16} + 1648981055585327 T^{17} + 8443403994806727 T^{18} + 40675585999222302 T^{19} + 194198291880554721 T^{20} + 872310978404637983 T^{21} + 3886927241494749037 T^{22} + 16230875597882775674 T^{23} + 67452262990443595171 T^{24} +$$$$26\!\cdots\!73$$$$T^{25} +$$$$10\!\cdots\!21$$$$T^{26} +$$$$35\!\cdots\!88$$$$T^{27} +$$$$12\!\cdots\!61$$$$T^{28} +$$$$40\!\cdots\!11$$$$T^{29} +$$$$13\!\cdots\!75$$$$T^{30} +$$$$37\!\cdots\!06$$$$T^{31} +$$$$11\!\cdots\!36$$$$T^{32} +$$$$26\!\cdots\!51$$$$T^{33} +$$$$73\!\cdots\!51$$$$T^{34} +$$$$12\!\cdots\!29$$$$T^{35} +$$$$32\!\cdots\!90$$$$T^{36} +$$$$32\!\cdots\!90$$$$T^{37} +$$$$74\!\cdots\!87$$$$T^{38}$$)
$29$ ($$1 + 7 T + 78 T^{2} + 6 T^{3} - 100 T^{4} - 14311 T^{5} - 2900 T^{6} + 5046 T^{7} + 1902342 T^{8} + 4950967 T^{9} + 20511149 T^{10}$$)($$1 + T + 165 T^{2} + 133 T^{3} + 11578 T^{4} + 7373 T^{5} + 442770 T^{6} + 213817 T^{7} + 9737098 T^{8} + 3243737 T^{9} + 116701365 T^{10} + 20511149 T^{11} + 594823321 T^{12}$$)($$1 - 27 T + 543 T^{2} - 7731 T^{3} + 94409 T^{4} - 972228 T^{5} + 9099203 T^{6} - 76790998 T^{7} + 609906611 T^{8} - 4538969769 T^{9} + 32519557765 T^{10} - 223195611792 T^{11} + 1489413751209 T^{12} - 9601114071190 T^{13} + 60213588863043 T^{14} - 365598701408931 T^{15} + 2159271329728813 T^{16} - 12384125184864807 T^{17} + 69251480492257511 T^{18} - 377365268591971366 T^{19} + 2008292934275467819 T^{20} - 10415049280471302687 T^{21} + 52662468460756020257 T^{22} -$$$$25\!\cdots\!11$$$$T^{23} +$$$$12\!\cdots\!07$$$$T^{24} -$$$$57\!\cdots\!90$$$$T^{25} +$$$$25\!\cdots\!81$$$$T^{26} -$$$$11\!\cdots\!12$$$$T^{27} +$$$$47\!\cdots\!85$$$$T^{28} -$$$$19\!\cdots\!69$$$$T^{29} +$$$$74\!\cdots\!19$$$$T^{30} -$$$$27\!\cdots\!18$$$$T^{31} +$$$$93\!\cdots\!67$$$$T^{32} -$$$$28\!\cdots\!68$$$$T^{33} +$$$$81\!\cdots\!41$$$$T^{34} -$$$$19\!\cdots\!51$$$$T^{35} +$$$$39\!\cdots\!87$$$$T^{36} -$$$$56\!\cdots\!47$$$$T^{37} +$$$$61\!\cdots\!69$$$$T^{38}$$)
$31$ ($$1 + 13 T + 163 T^{2} + 1437 T^{3} + 10393 T^{4} + 64421 T^{5} + 322183 T^{6} + 1380957 T^{7} + 4855933 T^{8} + 12005773 T^{9} + 28629151 T^{10}$$)($$1 + 10 T + 211 T^{2} + 1529 T^{3} + 17373 T^{4} + 94023 T^{5} + 731173 T^{6} + 2914713 T^{7} + 16695453 T^{8} + 45550439 T^{9} + 194862931 T^{10} + 286291510 T^{11} + 887503681 T^{12}$$)($$1 - 7 T + 313 T^{2} - 1745 T^{3} + 48034 T^{4} - 228546 T^{5} + 4976968 T^{6} - 21203546 T^{7} + 393833617 T^{8} - 1547203197 T^{9} + 25251760587 T^{10} - 93132965616 T^{11} + 1358073204047 T^{12} - 4750647003895 T^{13} + 62639119135757 T^{14} - 208507084444934 T^{15} + 2512919669470499 T^{16} - 7940015937146499 T^{17} + 88435740818992781 T^{18} - 263486053627797950 T^{19} + 2741507965388776211 T^{20} - 7630355315597785539 T^{21} + 74862389873195635709 T^{22} -$$$$19\!\cdots\!14$$$$T^{23} +$$$$17\!\cdots\!07$$$$T^{24} -$$$$42\!\cdots\!95$$$$T^{25} +$$$$37\!\cdots\!17$$$$T^{26} -$$$$79\!\cdots\!56$$$$T^{27} +$$$$66\!\cdots\!77$$$$T^{28} -$$$$12\!\cdots\!97$$$$T^{29} +$$$$10\!\cdots\!27$$$$T^{30} -$$$$16\!\cdots\!06$$$$T^{31} +$$$$12\!\cdots\!88$$$$T^{32} -$$$$17\!\cdots\!66$$$$T^{33} +$$$$11\!\cdots\!34$$$$T^{34} -$$$$12\!\cdots\!45$$$$T^{35} +$$$$70\!\cdots\!43$$$$T^{36} -$$$$48\!\cdots\!87$$$$T^{37} +$$$$21\!\cdots\!71$$$$T^{38}$$)
$37$ ($$1 + 6 T + 153 T^{2} + 749 T^{3} + 10446 T^{4} + 39095 T^{5} + 386502 T^{6} + 1025381 T^{7} + 7749909 T^{8} + 11244966 T^{9} + 69343957 T^{10}$$)($$1 + 19 T + 254 T^{2} + 2282 T^{3} + 17160 T^{4} + 108036 T^{5} + 672701 T^{6} + 3997332 T^{7} + 23492040 T^{8} + 115590146 T^{9} + 476036894 T^{10} + 1317535183 T^{11} + 2565726409 T^{12}$$)($$1 - 20 T + 529 T^{2} - 7579 T^{3} + 121179 T^{4} - 1399089 T^{5} + 17203375 T^{6} - 170179590 T^{7} + 1761811754 T^{8} - 15480072247 T^{9} + 141219691014 T^{10} - 1126765129536 T^{11} + 9296912563794 T^{12} - 68317644444629 T^{13} + 518158068049246 T^{14} - 3539692761737466 T^{15} + 24938736417249380 T^{16} - 159309392255327519 T^{17} + 1049317852548917248 T^{18} - 6285947908952657866 T^{19} + 38824760544309938176 T^{20} -$$$$21\!\cdots\!11$$$$T^{21} +$$$$12\!\cdots\!40$$$$T^{22} -$$$$66\!\cdots\!26$$$$T^{23} +$$$$35\!\cdots\!22$$$$T^{24} -$$$$17\!\cdots\!61$$$$T^{25} +$$$$88\!\cdots\!02$$$$T^{26} -$$$$39\!\cdots\!56$$$$T^{27} +$$$$18\!\cdots\!78$$$$T^{28} -$$$$74\!\cdots\!03$$$$T^{29} +$$$$31\!\cdots\!02$$$$T^{30} -$$$$11\!\cdots\!90$$$$T^{31} +$$$$41\!\cdots\!75$$$$T^{32} -$$$$12\!\cdots\!21$$$$T^{33} +$$$$40\!\cdots\!47$$$$T^{34} -$$$$93\!\cdots\!39$$$$T^{35} +$$$$24\!\cdots\!93$$$$T^{36} -$$$$33\!\cdots\!80$$$$T^{37} +$$$$62\!\cdots\!73$$$$T^{38}$$)
$41$ ($$1 + 9 T + 206 T^{2} + 1354 T^{3} + 16734 T^{4} + 80727 T^{5} + 686094 T^{6} + 2276074 T^{7} + 14197726 T^{8} + 25431849 T^{9} + 115856201 T^{10}$$)($$1 + 7 T + 157 T^{2} + 1231 T^{3} + 11116 T^{4} + 93909 T^{5} + 522244 T^{6} + 3850269 T^{7} + 18685996 T^{8} + 84841751 T^{9} + 443644477 T^{10} + 810993407 T^{11} + 4750104241 T^{12}$$)($$1 - 19 T + 506 T^{2} - 6636 T^{3} + 107341 T^{4} - 1105926 T^{5} + 13788708 T^{6} - 118113822 T^{7} + 1248252525 T^{8} - 9179930965 T^{9} + 87252857649 T^{10} - 564463578812 T^{11} + 5060173753283 T^{12} - 29512057711503 T^{13} + 258914623693343 T^{14} - 1396951401567362 T^{15} + 12193461557429751 T^{16} - 62175477043936019 T^{17} + 536335147097237381 T^{18} - 2622017764078053616 T^{19} + 21989741030986732621 T^{20} -$$$$10\!\cdots\!39$$$$T^{21} +$$$$84\!\cdots\!71$$$$T^{22} -$$$$39\!\cdots\!82$$$$T^{23} +$$$$29\!\cdots\!43$$$$T^{24} -$$$$14\!\cdots\!23$$$$T^{25} +$$$$98\!\cdots\!23$$$$T^{26} -$$$$45\!\cdots\!52$$$$T^{27} +$$$$28\!\cdots\!89$$$$T^{28} -$$$$12\!\cdots\!65$$$$T^{29} +$$$$68\!\cdots\!25$$$$T^{30} -$$$$26\!\cdots\!82$$$$T^{31} +$$$$12\!\cdots\!68$$$$T^{32} -$$$$41\!\cdots\!86$$$$T^{33} +$$$$16\!\cdots\!41$$$$T^{34} -$$$$42\!\cdots\!76$$$$T^{35} +$$$$13\!\cdots\!86$$$$T^{36} -$$$$20\!\cdots\!99$$$$T^{37} +$$$$43\!\cdots\!61$$$$T^{38}$$)
$43$ ($$1 - 2 T + 149 T^{2} - 97 T^{3} + 10204 T^{4} - 1983 T^{5} + 438772 T^{6} - 179353 T^{7} + 11846543 T^{8} - 6837602 T^{9} + 147008443 T^{10}$$)($$1 + 2 T + 184 T^{2} + 585 T^{3} + 15115 T^{4} + 56676 T^{5} + 778470 T^{6} + 2437068 T^{7} + 27947635 T^{8} + 46511595 T^{9} + 629059384 T^{10} + 294016886 T^{11} + 6321363049 T^{12}$$)($$1 - 20 T + 572 T^{2} - 8077 T^{3} + 135512 T^{4} - 1497988 T^{5} + 18790148 T^{6} - 171851865 T^{7} + 1774200657 T^{8} - 13927589453 T^{9} + 125276923181 T^{10} - 868765034041 T^{11} + 7102666891208 T^{12} - 44614823605178 T^{13} + 343538447765116 T^{14} - 2004023717345733 T^{15} + 15033040387500544 T^{16} - 84102416638225978 T^{17} + 634888635999058653 T^{18} - 3543965150345565646 T^{19} + 27300211347959522079 T^{20} -$$$$15\!\cdots\!22$$$$T^{21} +$$$$11\!\cdots\!08$$$$T^{22} -$$$$68\!\cdots\!33$$$$T^{23} +$$$$50\!\cdots\!88$$$$T^{24} -$$$$28\!\cdots\!22$$$$T^{25} +$$$$19\!\cdots\!56$$$$T^{26} -$$$$10\!\cdots\!41$$$$T^{27} +$$$$62\!\cdots\!83$$$$T^{28} -$$$$30\!\cdots\!97$$$$T^{29} +$$$$16\!\cdots\!99$$$$T^{30} -$$$$68\!\cdots\!65$$$$T^{31} +$$$$32\!\cdots\!64$$$$T^{32} -$$$$11\!\cdots\!12$$$$T^{33} +$$$$43\!\cdots\!84$$$$T^{34} -$$$$11\!\cdots\!77$$$$T^{35} +$$$$33\!\cdots\!96$$$$T^{36} -$$$$50\!\cdots\!80$$$$T^{37} +$$$$10\!\cdots\!07$$$$T^{38}$$)
$47$ ($$1 + 3 T + 213 T^{2} + 457 T^{3} + 18859 T^{4} + 29687 T^{5} + 886373 T^{6} + 1009513 T^{7} + 22114299 T^{8} + 14639043 T^{9} + 229345007 T^{10}$$)($$1 - 5 T + 172 T^{2} - 706 T^{3} + 15414 T^{4} - 52066 T^{5} + 869866 T^{6} - 2447102 T^{7} + 34049526 T^{8} - 73299038 T^{9} + 839305132 T^{10} - 1146725035 T^{11} + 10779215329 T^{12}$$)($$1 + 19 T + 486 T^{2} + 6712 T^{3} + 102965 T^{4} + 1146923 T^{5} + 13578502 T^{6} + 130762400 T^{7} + 1326239937 T^{8} + 11621437031 T^{9} + 106840792116 T^{10} + 879080625727 T^{11} + 7526347922476 T^{12} + 58855151532759 T^{13} + 474007528803564 T^{14} + 3531745422259027 T^{15} + 26901966093952584 T^{16} + 191280615501341947 T^{17} + 1385568946871706288 T^{18} + 9415498032408347691 T^{19} + 65121740502970195536 T^{20} +$$$$42\!\cdots\!23$$$$T^{21} +$$$$27\!\cdots\!32$$$$T^{22} +$$$$17\!\cdots\!87$$$$T^{23} +$$$$10\!\cdots\!48$$$$T^{24} +$$$$63\!\cdots\!11$$$$T^{25} +$$$$38\!\cdots\!88$$$$T^{26} +$$$$20\!\cdots\!47$$$$T^{27} +$$$$11\!\cdots\!72$$$$T^{28} +$$$$61\!\cdots\!19$$$$T^{29} +$$$$32\!\cdots\!11$$$$T^{30} +$$$$15\!\cdots\!00$$$$T^{31} +$$$$74\!\cdots\!54$$$$T^{32} +$$$$29\!\cdots\!87$$$$T^{33} +$$$$12\!\cdots\!95$$$$T^{34} +$$$$38\!\cdots\!52$$$$T^{35} +$$$$12\!\cdots\!82$$$$T^{36} +$$$$23\!\cdots\!91$$$$T^{37} +$$$$58\!\cdots\!83$$$$T^{38}$$)
$53$ ($$1 - 3 T + 135 T^{2} - 607 T^{3} + 10497 T^{4} - 45141 T^{5} + 556341 T^{6} - 1705063 T^{7} + 20098395 T^{8} - 23671443 T^{9} + 418195493 T^{10}$$)($$1 + 9 T + 236 T^{2} + 1806 T^{3} + 26894 T^{4} + 167050 T^{5} + 1820956 T^{6} + 8853650 T^{7} + 75545246 T^{8} + 268871862 T^{9} + 1862153516 T^{10} + 3763759437 T^{11} + 22164361129 T^{12}$$)($$1 - 3 T + 553 T^{2} - 1479 T^{3} + 152698 T^{4} - 391018 T^{5} + 28234280 T^{6} - 72193850 T^{7} + 3938669069 T^{8} - 10194471857 T^{9} + 441688867147 T^{10} - 1151757712728 T^{11} + 41342815285789 T^{12} - 107148447751419 T^{13} + 3305250065811811 T^{14} - 8375541396988102 T^{15} + 228888609419105435 T^{16} - 557600895340413919 T^{17} + 13840952306811604433 T^{18} - 31860554916814458306 T^{19} +$$$$73\!\cdots\!49$$$$T^{20} -$$$$15\!\cdots\!71$$$$T^{21} +$$$$34\!\cdots\!95$$$$T^{22} -$$$$66\!\cdots\!62$$$$T^{23} +$$$$13\!\cdots\!23$$$$T^{24} -$$$$23\!\cdots\!51$$$$T^{25} +$$$$48\!\cdots\!93$$$$T^{26} -$$$$71\!\cdots\!08$$$$T^{27} +$$$$14\!\cdots\!51$$$$T^{28} -$$$$17\!\cdots\!93$$$$T^{29} +$$$$36\!\cdots\!93$$$$T^{30} -$$$$35\!\cdots\!50$$$$T^{31} +$$$$73\!\cdots\!40$$$$T^{32} -$$$$53\!\cdots\!42$$$$T^{33} +$$$$11\!\cdots\!86$$$$T^{34} -$$$$57\!\cdots\!59$$$$T^{35} +$$$$11\!\cdots\!89$$$$T^{36} -$$$$32\!\cdots\!67$$$$T^{37} +$$$$57\!\cdots\!17$$$$T^{38}$$)
$59$ ($$1 + 14 T + 322 T^{2} + 2995 T^{3} + 38600 T^{4} + 256597 T^{5} + 2277400 T^{6} + 10425595 T^{7} + 66132038 T^{8} + 169643054 T^{9} + 714924299 T^{10}$$)($$1 + 5 T + 295 T^{2} + 1301 T^{3} + 39467 T^{4} + 144034 T^{5} + 3009281 T^{6} + 8498006 T^{7} + 137384627 T^{8} + 267198079 T^{9} + 3574621495 T^{10} + 3574621495 T^{11} + 42180533641 T^{12}$$)($$1 + 28 T + 710 T^{2} + 12043 T^{3} + 193793 T^{4} + 2523471 T^{5} + 31692672 T^{6} + 342644362 T^{7} + 3627986625 T^{8} + 33994128649 T^{9} + 317195915191 T^{10} + 2654252058056 T^{11} + 22606553903935 T^{12} + 173800732721867 T^{13} + 1400519610931769 T^{14} + 10218195396510582 T^{15} + 80843054411446731 T^{16} + 579240599508755665 T^{17} + 4631677843792297891 T^{18} + 33493722583958564698 T^{19} +$$$$27\!\cdots\!69$$$$T^{20} +$$$$20\!\cdots\!65$$$$T^{21} +$$$$16\!\cdots\!49$$$$T^{22} +$$$$12\!\cdots\!02$$$$T^{23} +$$$$10\!\cdots\!31$$$$T^{24} +$$$$73\!\cdots\!47$$$$T^{25} +$$$$56\!\cdots\!65$$$$T^{26} +$$$$38\!\cdots\!76$$$$T^{27} +$$$$27\!\cdots\!49$$$$T^{28} +$$$$17\!\cdots\!49$$$$T^{29} +$$$$10\!\cdots\!75$$$$T^{30} +$$$$60\!\cdots\!22$$$$T^{31} +$$$$33\!\cdots\!88$$$$T^{32} +$$$$15\!\cdots\!31$$$$T^{33} +$$$$70\!\cdots\!07$$$$T^{34} +$$$$25\!\cdots\!63$$$$T^{35} +$$$$90\!\cdots\!90$$$$T^{36} +$$$$21\!\cdots\!88$$$$T^{37} +$$$$44\!\cdots\!39$$$$T^{38}$$)
$61$ ($$( 1 - T )^{5}$$)($$( 1 + T )^{6}$$)($$( 1 + T )^{19}$$)
$67$ ($$1 + 5 T + 249 T^{2} + 870 T^{3} + 27449 T^{4} + 72939 T^{5} + 1839083 T^{6} + 3905430 T^{7} + 74889987 T^{8} + 100755605 T^{9} + 1350125107 T^{10}$$)($$1 + 14 T + 345 T^{2} + 2734 T^{3} + 39586 T^{4} + 203614 T^{5} + 2781137 T^{6} + 13642138 T^{7} + 177701554 T^{8} + 822286042 T^{9} + 6952136745 T^{10} + 18901751498 T^{11} + 90458382169 T^{12}$$)($$1 - 3 T + 617 T^{2} - 1296 T^{3} + 195240 T^{4} - 274406 T^{5} + 42305846 T^{6} - 35902906 T^{7} + 7029083151 T^{8} - 2751947733 T^{9} + 949648979305 T^{10} - 21362463620 T^{11} + 108075690044113 T^{12} + 27914267730953 T^{13} + 10597462643039983 T^{14} + 4749868698195514 T^{15} + 908526944174305233 T^{16} + 495817426944109421 T^{17} + 68710270553343799099 T^{18} + 38074969022944131304 T^{19} +$$$$46\!\cdots\!33$$$$T^{20} +$$$$22\!\cdots\!69$$$$T^{21} +$$$$27\!\cdots\!79$$$$T^{22} +$$$$95\!\cdots\!94$$$$T^{23} +$$$$14\!\cdots\!81$$$$T^{24} +$$$$25\!\cdots\!57$$$$T^{25} +$$$$65\!\cdots\!99$$$$T^{26} -$$$$86\!\cdots\!20$$$$T^{27} +$$$$25\!\cdots\!35$$$$T^{28} -$$$$50\!\cdots\!17$$$$T^{29} +$$$$85\!\cdots\!33$$$$T^{30} -$$$$29\!\cdots\!66$$$$T^{31} +$$$$23\!\cdots\!02$$$$T^{32} -$$$$10\!\cdots\!74$$$$T^{33} +$$$$48\!\cdots\!20$$$$T^{34} -$$$$21\!\cdots\!76$$$$T^{35} +$$$$68\!\cdots\!59$$$$T^{36} -$$$$22\!\cdots\!27$$$$T^{37} +$$$$49\!\cdots\!03$$$$T^{38}$$)
$71$ ($$1 - 3 T + 282 T^{2} - 1111 T^{3} + 34521 T^{4} - 127665 T^{5} + 2450991 T^{6} - 5600551 T^{7} + 100930902 T^{8} - 76235043 T^{9} + 1804229351 T^{10}$$)($$1 + 14 T + 370 T^{2} + 4244 T^{3} + 60622 T^{4} + 557781 T^{5} + 5582929 T^{6} + 39602451 T^{7} + 305595502 T^{8} + 1518974284 T^{9} + 9402321970 T^{10} + 25259210914 T^{11} + 128100283921 T^{12}$$)($$1 + 19 T + 878 T^{2} + 14241 T^{3} + 374920 T^{4} + 5352616 T^{5} + 104752181 T^{6} + 1343370758 T^{7} + 21623226530 T^{8} + 252588529195 T^{9} + 3518873427046 T^{10} + 37800903314792 T^{11} + 469312958702142 T^{12} + 4665621917976129 T^{13} + 52560369889705106 T^{14} + 485419890861740250 T^{15} + 5019201466897587468 T^{16} + 43139697670195372617 T^{17} +$$$$41\!\cdots\!54$$$$T^{18} +$$$$32\!\cdots\!34$$$$T^{19} +$$$$29\!\cdots\!34$$$$T^{20} +$$$$21\!\cdots\!97$$$$T^{21} +$$$$17\!\cdots\!48$$$$T^{22} +$$$$12\!\cdots\!50$$$$T^{23} +$$$$94\!\cdots\!06$$$$T^{24} +$$$$59\!\cdots\!09$$$$T^{25} +$$$$42\!\cdots\!22$$$$T^{26} +$$$$24\!\cdots\!12$$$$T^{27} +$$$$16\!\cdots\!26$$$$T^{28} +$$$$82\!\cdots\!95$$$$T^{29} +$$$$49\!\cdots\!30$$$$T^{30} +$$$$22\!\cdots\!78$$$$T^{31} +$$$$12\!\cdots\!91$$$$T^{32} +$$$$44\!\cdots\!96$$$$T^{33} +$$$$22\!\cdots\!20$$$$T^{34} +$$$$59\!\cdots\!61$$$$T^{35} +$$$$25\!\cdots\!98$$$$T^{36} +$$$$39\!\cdots\!59$$$$T^{37} +$$$$14\!\cdots\!31$$$$T^{38}$$)
$73$ ($$1 + 4 T + 247 T^{2} + 1069 T^{3} + 28266 T^{4} + 113873 T^{5} + 2063418 T^{6} + 5696701 T^{7} + 96087199 T^{8} + 113592964 T^{9} + 2073071593 T^{10}$$)($$1 + 14 T + 238 T^{2} + 1833 T^{3} + 26783 T^{4} + 214906 T^{5} + 2618754 T^{6} + 15688138 T^{7} + 142726607 T^{8} + 713068161 T^{9} + 6758781358 T^{10} + 29023002302 T^{11} + 151334226289 T^{12}$$)($$1 - 20 T + 983 T^{2} - 15447 T^{3} + 442119 T^{4} - 5776019 T^{5} + 124649027 T^{6} - 1395944110 T^{7} + 25117566350 T^{8} - 245527957621 T^{9} + 3892497812088 T^{10} - 33616760931648 T^{11} + 487253061211176 T^{12} - 3757469846070647 T^{13} + 51174778307022026 T^{14} - 357004941641166770 T^{15} + 4656335452940103068 T^{16} - 29922965328739857229 T^{17} +$$$$37\!\cdots\!70$$$$T^{18} -$$$$22\!\cdots\!34$$$$T^{19} +$$$$27\!\cdots\!10$$$$T^{20} -$$$$15\!\cdots\!41$$$$T^{21} +$$$$18\!\cdots\!56$$$$T^{22} -$$$$10\!\cdots\!70$$$$T^{23} +$$$$10\!\cdots\!18$$$$T^{24} -$$$$56\!\cdots\!83$$$$T^{25} +$$$$53\!\cdots\!72$$$$T^{26} -$$$$27\!\cdots\!88$$$$T^{27} +$$$$22\!\cdots\!44$$$$T^{28} -$$$$10\!\cdots\!29$$$$T^{29} +$$$$78\!\cdots\!50$$$$T^{30} -$$$$31\!\cdots\!10$$$$T^{31} +$$$$20\!\cdots\!91$$$$T^{32} -$$$$70\!\cdots\!71$$$$T^{33} +$$$$39\!\cdots\!83$$$$T^{34} -$$$$10\!\cdots\!67$$$$T^{35} +$$$$46\!\cdots\!99$$$$T^{36} -$$$$69\!\cdots\!80$$$$T^{37} +$$$$25\!\cdots\!37$$$$T^{38}$$)
$79$ ($$1 + 27 T + 654 T^{2} + 9608 T^{3} + 125752 T^{4} + 1182335 T^{5} + 9934408 T^{6} + 59963528 T^{7} + 322447506 T^{8} + 1051652187 T^{9} + 3077056399 T^{10}$$)($$1 - 5 T + 311 T^{2} - 1249 T^{3} + 49564 T^{4} - 164781 T^{5} + 4854524 T^{6} - 13017699 T^{7} + 309328924 T^{8} - 615805711 T^{9} + 12113475191 T^{10} - 15385281995 T^{11} + 243087455521 T^{12}$$)($$1 - 69 T + 2783 T^{2} - 80271 T^{3} + 1833295 T^{4} - 34810424 T^{5} + 569306503 T^{6} - 8213895354 T^{7} + 106756573323 T^{8} - 1272070190155 T^{9} + 14143909234770 T^{10} - 149273308631457 T^{11} + 1522063396688776 T^{12} - 15229257243053839 T^{13} + 151241951360093818 T^{14} - 1495740235635165111 T^{15} + 14669808097880173198 T^{16} -$$$$14\!\cdots\!01$$$$T^{17} +$$$$13\!\cdots\!22$$$$T^{18} -$$$$11\!\cdots\!37$$$$T^{19} +$$$$10\!\cdots\!38$$$$T^{20} -$$$$88\!\cdots\!41$$$$T^{21} +$$$$72\!\cdots\!22$$$$T^{22} -$$$$58\!\cdots\!91$$$$T^{23} +$$$$46\!\cdots\!82$$$$T^{24} -$$$$37\!\cdots\!19$$$$T^{25} +$$$$29\!\cdots\!84$$$$T^{26} -$$$$22\!\cdots\!77$$$$T^{27} +$$$$16\!\cdots\!30$$$$T^{28} -$$$$12\!\cdots\!55$$$$T^{29} +$$$$79\!\cdots\!17$$$$T^{30} -$$$$48\!\cdots\!14$$$$T^{31} +$$$$26\!\cdots\!17$$$$T^{32} -$$$$12\!\cdots\!44$$$$T^{33} +$$$$53\!\cdots\!05$$$$T^{34} -$$$$18\!\cdots\!91$$$$T^{35} +$$$$50\!\cdots\!97$$$$T^{36} -$$$$99\!\cdots\!09$$$$T^{37} +$$$$11\!\cdots\!19$$$$T^{38}$$)
$83$ ($$1 + 3 T + 272 T^{2} + 278 T^{3} + 34402 T^{4} + 8271 T^{5} + 2855366 T^{6} + 1915142 T^{7} + 155526064 T^{8} + 142374963 T^{9} + 3939040643 T^{10}$$)($$1 - 17 T + 327 T^{2} - 4281 T^{3} + 55034 T^{4} - 551657 T^{5} + 5713940 T^{6} - 45787531 T^{7} + 379129226 T^{8} - 2447820147 T^{9} + 15518870967 T^{10} - 66963690931 T^{11} + 326940373369 T^{12}$$)($$1 - T + 546 T^{2} + 622 T^{3} + 151951 T^{4} + 457032 T^{5} + 29392894 T^{6} + 133790922 T^{7} + 4497136511 T^{8} + 25906914375 T^{9} + 583122186185 T^{10} + 3883388716012 T^{11} + 66936907499529 T^{12} + 486267239143469 T^{13} + 6998628514481439 T^{14} + 53000991615274038 T^{15} + 676682691789200105 T^{16} + 5139164953933873773 T^{17} + 60700141269974555373 T^{18} +$$$$44\!\cdots\!16$$$$T^{19} +$$$$50\!\cdots\!59$$$$T^{20} +$$$$35\!\cdots\!97$$$$T^{21} +$$$$38\!\cdots\!35$$$$T^{22} +$$$$25\!\cdots\!98$$$$T^{23} +$$$$27\!\cdots\!77$$$$T^{24} +$$$$15\!\cdots\!61$$$$T^{25} +$$$$18\!\cdots\!83$$$$T^{26} +$$$$87\!\cdots\!92$$$$T^{27} +$$$$10\!\cdots\!55$$$$T^{28} +$$$$40\!\cdots\!75$$$$T^{29} +$$$$57\!\cdots\!37$$$$T^{30} +$$$$14\!\cdots\!42$$$$T^{31} +$$$$26\!\cdots\!22$$$$T^{32} +$$$$33\!\cdots\!28$$$$T^{33} +$$$$92\!\cdots\!57$$$$T^{34} +$$$$31\!\cdots\!82$$$$T^{35} +$$$$22\!\cdots\!58$$$$T^{36} -$$$$34\!\cdots\!09$$$$T^{37} +$$$$29\!\cdots\!47$$$$T^{38}$$)
$89$ ($$1 + 12 T + 370 T^{2} + 3531 T^{3} + 58932 T^{4} + 442409 T^{5} + 5244948 T^{6} + 27969051 T^{7} + 260838530 T^{8} + 752906892 T^{9} + 5584059449 T^{10}$$)($$1 + 25 T + 649 T^{2} + 10981 T^{3} + 159015 T^{4} + 1943100 T^{5} + 19432603 T^{6} + 172935900 T^{7} + 1259557815 T^{8} + 7741264589 T^{9} + 40719714409 T^{10} + 139601486225 T^{11} + 496981290961 T^{12}$$)($$1 + 712 T^{2} + 987 T^{3} + 245239 T^{4} + 620559 T^{5} + 55410092 T^{6} + 193333154 T^{7} + 9368692181 T^{8} + 41204661989 T^{9} + 1279635571133 T^{10} + 6988180468952 T^{11} + 149026463314535 T^{12} + 1016065923599399 T^{13} + 15504500517386055 T^{14} + 129196593580182910 T^{15} + 1501138729702439567 T^{16} + 14299173695878752965 T^{17} +$$$$13\!\cdots\!37$$$$T^{18} +$$$$13\!\cdots\!42$$$$T^{19} +$$$$12\!\cdots\!93$$$$T^{20} +$$$$11\!\cdots\!65$$$$T^{21} +$$$$10\!\cdots\!23$$$$T^{22} +$$$$81\!\cdots\!10$$$$T^{23} +$$$$86\!\cdots\!95$$$$T^{24} +$$$$50\!\cdots\!39$$$$T^{25} +$$$$65\!\cdots\!15$$$$T^{26} +$$$$27\!\cdots\!12$$$$T^{27} +$$$$44\!\cdots\!97$$$$T^{28} +$$$$12\!\cdots\!89$$$$T^{29} +$$$$25\!\cdots\!09$$$$T^{30} +$$$$47\!\cdots\!34$$$$T^{31} +$$$$12\!\cdots\!48$$$$T^{32} +$$$$12\!\cdots\!19$$$$T^{33} +$$$$42\!\cdots\!11$$$$T^{34} +$$$$15\!\cdots\!07$$$$T^{35} +$$$$98\!\cdots\!48$$$$T^{36} +$$$$10\!\cdots\!09$$$$T^{38}$$)
$97$ ($$1 + 5 T + 258 T^{2} + 578 T^{3} + 37892 T^{4} + 80189 T^{5} + 3675524 T^{6} + 5438402 T^{7} + 235469634 T^{8} + 442646405 T^{9} + 8587340257 T^{10}$$)($$1 + 24 T + 678 T^{2} + 10373 T^{3} + 170592 T^{4} + 1896695 T^{5} + 22185859 T^{6} + 183979415 T^{7} + 1605100128 T^{8} + 9467157029 T^{9} + 60022852518 T^{10} + 206096166168 T^{11} + 832972004929 T^{12}$$)($$1 - 21 T + 847 T^{2} - 11527 T^{3} + 298607 T^{4} - 2986298 T^{5} + 68446913 T^{6} - 534920010 T^{7} + 12571255465 T^{8} - 79993220799 T^{9} + 2032794671226 T^{10} - 10864472031073 T^{11} + 294914282878728 T^{12} - 1337284301131743 T^{13} + 38226363126660730 T^{14} - 149144666063131807 T^{15} + 4466890173200617866 T^{16} - 15529883283856504905 T^{17} +$$$$47\!\cdots\!92$$$$T^{18} -$$$$15\!\cdots\!23$$$$T^{19} +$$$$46\!\cdots\!24$$$$T^{20} -$$$$14\!\cdots\!45$$$$T^{21} +$$$$40\!\cdots\!18$$$$T^{22} -$$$$13\!\cdots\!67$$$$T^{23} +$$$$32\!\cdots\!10$$$$T^{24} -$$$$11\!\cdots\!47$$$$T^{25} +$$$$23\!\cdots\!64$$$$T^{26} -$$$$85\!\cdots\!53$$$$T^{27} +$$$$15\!\cdots\!42$$$$T^{28} -$$$$58\!\cdots\!51$$$$T^{29} +$$$$89\!\cdots\!45$$$$T^{30} -$$$$37\!\cdots\!10$$$$T^{31} +$$$$46\!\cdots\!01$$$$T^{32} -$$$$19\!\cdots\!62$$$$T^{33} +$$$$18\!\cdots\!51$$$$T^{34} -$$$$70\!\cdots\!67$$$$T^{35} +$$$$50\!\cdots\!39$$$$T^{36} -$$$$12\!\cdots\!69$$$$T^{37} +$$$$56\!\cdots\!33$$$$T^{38}$$)