Properties

Label 693.2.m
Level 693
Weight 2
Character orbit m
Rep. character \(\chi_{693}(64,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 120
Newform subspaces 11
Sturm bound 192
Trace bound 2

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Defining parameters

Level: \( N \) = \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 693.m (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 11 \)
Sturm bound: \(192\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(693, [\chi])\).

Total New Old
Modular forms 416 120 296
Cusp forms 352 120 232
Eisenstein series 64 0 64

Trace form

\( 120q - 32q^{4} + 2q^{5} - 2q^{7} + 10q^{8} + O(q^{10}) \) \( 120q - 32q^{4} + 2q^{5} - 2q^{7} + 10q^{8} + 16q^{10} + 10q^{11} + 14q^{13} + 3q^{14} - 50q^{16} + 8q^{17} + 26q^{19} + 38q^{20} - 6q^{22} - 60q^{25} + 58q^{26} - q^{28} - 10q^{29} - 10q^{31} - 100q^{32} - 16q^{34} + 8q^{35} + 2q^{37} - 34q^{38} + 70q^{40} + 24q^{41} + 24q^{43} - 9q^{44} + 27q^{46} + 24q^{47} - 30q^{49} + 46q^{50} - 26q^{52} + 46q^{53} - 42q^{55} - 18q^{56} - 51q^{58} - 38q^{59} - 106q^{62} - 72q^{64} - 56q^{65} + 44q^{67} - 94q^{68} + 20q^{70} + 24q^{71} - 64q^{73} + 66q^{74} - 112q^{76} - 8q^{77} - 66q^{79} - 56q^{80} - 86q^{82} + 60q^{83} - 12q^{85} - 33q^{86} + 84q^{88} + 24q^{89} + 18q^{91} + 79q^{92} + 10q^{94} + 72q^{95} + 58q^{97} - 10q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(693, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
693.2.m.a \(4\) \(5.534\) \(\Q(\zeta_{10})\) None \(-5\) \(0\) \(3\) \(-1\) \(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
693.2.m.b \(4\) \(5.534\) \(\Q(\zeta_{10})\) None \(-2\) \(0\) \(1\) \(-1\) \(q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}+(-1+\zeta_{10}+\cdots)q^{4}+\cdots\)
693.2.m.c \(4\) \(5.534\) \(\Q(\zeta_{10})\) None \(-1\) \(0\) \(-7\) \(-1\) \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
693.2.m.d \(4\) \(5.534\) \(\Q(\zeta_{10})\) None \(2\) \(0\) \(-3\) \(1\) \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+(-1+\zeta_{10}+\zeta_{10}^{3})q^{4}+\cdots\)
693.2.m.e \(4\) \(5.534\) \(\Q(\zeta_{10})\) None \(4\) \(0\) \(3\) \(-1\) \(q+(2-\zeta_{10}+\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+(3+\cdots)q^{4}+\cdots\)
693.2.m.f \(8\) \(5.534\) 8.0.13140625.1 None \(-2\) \(0\) \(-2\) \(2\) \(q+(\beta _{3}-\beta _{5}-\beta _{6})q^{2}+(1+\beta _{4}+\beta _{6}+\cdots)q^{4}+\cdots\)
693.2.m.g \(8\) \(5.534\) 8.0.159390625.1 None \(1\) \(0\) \(-3\) \(2\) \(q-\beta _{4}q^{2}+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
693.2.m.h \(16\) \(5.534\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{1}q^{2}-\beta _{7}q^{4}+(\beta _{9}-\beta _{13})q^{5}-\beta _{4}q^{7}+\cdots\)
693.2.m.i \(16\) \(5.534\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(3\) \(0\) \(5\) \(-4\) \(q+(\beta _{5}+\beta _{6})q^{2}+(\beta _{2}+\beta _{7}+\beta _{11})q^{4}+\cdots\)
693.2.m.j \(20\) \(5.534\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(5\) \(5\) \(q-\beta _{7}q^{2}+(\beta _{3}-\beta _{6}-\beta _{8}-2\beta _{9}-\beta _{10}+\cdots)q^{4}+\cdots\)
693.2.m.k \(32\) \(5.534\) None \(0\) \(0\) \(0\) \(-8\)

Decomposition of \(S_{2}^{\mathrm{old}}(693, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(693, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(231, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 50 T^{5} + 52 T^{6} + 40 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 5 T^{3} + 11 T^{4} - 10 T^{5} + 8 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 - 4 T + 4 T^{2} + 7 T^{3} - 21 T^{4} + 14 T^{5} + 16 T^{6} - 32 T^{7} + 16 T^{8} \))(\( 1 + 2 T - 3 T^{2} - 6 T^{3} + 3 T^{4} + 6 T^{5} + 9 T^{6} + 2 T^{7} - 27 T^{8} + 4 T^{9} + 36 T^{10} + 48 T^{11} + 48 T^{12} - 192 T^{13} - 192 T^{14} + 256 T^{15} + 256 T^{16} \))(\( 1 - T - 3 T^{2} + 5 T^{3} - 19 T^{5} + 21 T^{6} + 20 T^{7} - 51 T^{8} + 40 T^{9} + 84 T^{10} - 152 T^{11} + 160 T^{13} - 192 T^{14} - 128 T^{15} + 256 T^{16} \))(\( 1 - T^{2} + T^{4} - 7 T^{8} - 10 T^{10} - 16 T^{12} + 74 T^{14} - 47 T^{16} + 296 T^{18} - 256 T^{20} - 640 T^{22} - 1792 T^{24} + 4096 T^{28} - 16384 T^{30} + 65536 T^{32} \))(\( 1 - 3 T + 6 T^{2} - 12 T^{3} + 18 T^{4} - 17 T^{5} + 15 T^{6} + T^{7} - 58 T^{8} + 119 T^{9} - 185 T^{10} + 292 T^{11} - 242 T^{12} - 33 T^{13} + 312 T^{14} - 878 T^{15} + 1785 T^{16} - 1756 T^{17} + 1248 T^{18} - 264 T^{19} - 3872 T^{20} + 9344 T^{21} - 11840 T^{22} + 15232 T^{23} - 14848 T^{24} + 512 T^{25} + 15360 T^{26} - 34816 T^{27} + 73728 T^{28} - 98304 T^{29} + 98304 T^{30} - 98304 T^{31} + 65536 T^{32} \))(\( 1 + 2 T^{2} - 3 T^{3} + 6 T^{4} + 8 T^{5} + 14 T^{6} - 5 T^{7} + 10 T^{8} - 2 T^{9} + 176 T^{10} - 67 T^{11} + 170 T^{12} - 190 T^{13} + 510 T^{14} + 495 T^{15} + 710 T^{16} - 492 T^{17} + 1850 T^{18} - 927 T^{19} + 8347 T^{20} - 1854 T^{21} + 7400 T^{22} - 3936 T^{23} + 11360 T^{24} + 15840 T^{25} + 32640 T^{26} - 24320 T^{27} + 43520 T^{28} - 34304 T^{29} + 180224 T^{30} - 4096 T^{31} + 40960 T^{32} - 40960 T^{33} + 229376 T^{34} + 262144 T^{35} + 393216 T^{36} - 393216 T^{37} + 524288 T^{38} + 1048576 T^{40} \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 15 T^{5} - 25 T^{6} - 375 T^{7} + 625 T^{8} \))(\( 1 - T + T^{2} - 11 T^{3} + 36 T^{4} - 55 T^{5} + 25 T^{6} - 125 T^{7} + 625 T^{8} \))(\( 1 + 7 T + 29 T^{2} + 93 T^{3} + 236 T^{4} + 465 T^{5} + 725 T^{6} + 875 T^{7} + 625 T^{8} \))(\( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 15 T^{5} - 25 T^{6} + 375 T^{7} + 625 T^{8} \))(\( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 15 T^{5} - 25 T^{6} - 375 T^{7} + 625 T^{8} \))(\( 1 + 2 T + 10 T^{2} + 42 T^{3} + 99 T^{4} + 338 T^{5} + 900 T^{6} + 2008 T^{7} + 5281 T^{8} + 10040 T^{9} + 22500 T^{10} + 42250 T^{11} + 61875 T^{12} + 131250 T^{13} + 156250 T^{14} + 156250 T^{15} + 390625 T^{16} \))(\( 1 + 3 T - 3 T^{2} - 30 T^{3} - 39 T^{4} + 150 T^{5} + 428 T^{6} - 231 T^{7} - 2169 T^{8} - 1155 T^{9} + 10700 T^{10} + 18750 T^{11} - 24375 T^{12} - 93750 T^{13} - 46875 T^{14} + 234375 T^{15} + 390625 T^{16} \))(\( ( 1 - 9 T^{2} + 21 T^{4} + 161 T^{6} - 1644 T^{8} + 4025 T^{10} + 13125 T^{12} - 140625 T^{14} + 390625 T^{16} )^{2} \))(\( 1 - 5 T - T^{2} + 66 T^{3} - 156 T^{4} - 267 T^{5} + 1789 T^{6} - 1241 T^{7} - 10249 T^{8} + 25094 T^{9} + 25181 T^{10} - 188691 T^{11} + 137316 T^{12} + 851484 T^{13} - 1968914 T^{14} - 1709452 T^{15} + 12417791 T^{16} - 8547260 T^{17} - 49222850 T^{18} + 106435500 T^{19} + 85822500 T^{20} - 589659375 T^{21} + 393453125 T^{22} + 1960468750 T^{23} - 4003515625 T^{24} - 2423828125 T^{25} + 17470703125 T^{26} - 13037109375 T^{27} - 38085937500 T^{28} + 80566406250 T^{29} - 6103515625 T^{30} - 152587890625 T^{31} + 152587890625 T^{32} \))(\( 1 - 5 T + 4 T^{2} + 32 T^{3} - 105 T^{4} + 10 T^{5} + 535 T^{6} - 956 T^{7} - 452 T^{8} + 4251 T^{9} - 8164 T^{10} + 5041 T^{11} + 36064 T^{12} - 140288 T^{13} + 128201 T^{14} + 579422 T^{15} - 1696447 T^{16} + 345724 T^{17} + 4805888 T^{18} - 5640991 T^{19} - 3186474 T^{20} - 28204955 T^{21} + 120147200 T^{22} + 43215500 T^{23} - 1060279375 T^{24} + 1810693750 T^{25} + 2003140625 T^{26} - 10960000000 T^{27} + 14087500000 T^{28} + 9845703125 T^{29} - 79726562500 T^{30} + 207568359375 T^{31} - 110351562500 T^{32} - 1166992187500 T^{33} + 3265380859375 T^{34} + 305175781250 T^{35} - 16021728515625 T^{36} + 24414062500000 T^{37} + 15258789062500 T^{38} - 95367431640625 T^{39} + 95367431640625 T^{40} \))
$7$ (\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5} \))
$11$ (\( 1 + T + 21 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 - T + 21 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 + T - 9 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + 9 T + 41 T^{2} + 99 T^{3} + 121 T^{4} \))(\( 1 + T - 9 T^{2} + 11 T^{3} + 121 T^{4} \))(\( ( 1 - 11 T + 51 T^{2} - 121 T^{3} + 121 T^{4} )^{2} \))(\( 1 + 5 T + 26 T^{2} + 75 T^{3} + 251 T^{4} + 825 T^{5} + 3146 T^{6} + 6655 T^{7} + 14641 T^{8} \))(\( 1 + 12 T^{2} + 293 T^{4} + 3444 T^{6} + 47125 T^{8} + 416724 T^{10} + 4289813 T^{12} + 21258732 T^{14} + 214358881 T^{16} \))(\( 1 - 3 T + 19 T^{2} + 76 T^{3} - 286 T^{4} + 2245 T^{5} + 1052 T^{6} - 7668 T^{7} + 114073 T^{8} - 84348 T^{9} + 127292 T^{10} + 2988095 T^{11} - 4187326 T^{12} + 12239876 T^{13} + 33659659 T^{14} - 58461513 T^{15} + 214358881 T^{16} \))(\( 1 - T - T^{2} - 25 T^{3} + 246 T^{4} - 17 T^{5} - 807 T^{6} - 1153 T^{7} + 25424 T^{8} + 22607 T^{9} - 174843 T^{10} + 248677 T^{11} + 3076304 T^{12} - 1534643 T^{13} - 11815287 T^{14} - 2737867 T^{15} + 435804006 T^{16} - 487179275 T^{17} - 214358881 T^{18} - 2357947691 T^{19} + 25937424601 T^{20} \))
$13$ (\( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 650 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8} \))(\( 1 - T - 12 T^{2} + 25 T^{3} + 131 T^{4} + 325 T^{5} - 2028 T^{6} - 2197 T^{7} + 28561 T^{8} \))(\( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 650 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8} \))(\( 1 + 7 T + 6 T^{2} - 49 T^{3} - 181 T^{4} - 637 T^{5} + 1014 T^{6} + 15379 T^{7} + 28561 T^{8} \))(\( 1 + 11 T + 48 T^{2} + 145 T^{3} + 491 T^{4} + 1885 T^{5} + 8112 T^{6} + 24167 T^{7} + 28561 T^{8} \))(\( 1 + 8 T - 6 T^{2} - 254 T^{3} - 569 T^{4} + 3664 T^{5} + 18260 T^{6} - 21760 T^{7} - 308143 T^{8} - 282880 T^{9} + 3085940 T^{10} + 8049808 T^{11} - 16251209 T^{12} - 94308422 T^{13} - 28960854 T^{14} + 501988136 T^{15} + 815730721 T^{16} \))(\( 1 - 5 T + 35 T^{2} - 90 T^{3} + 611 T^{4} - 670 T^{5} + 4380 T^{6} + 8865 T^{7} + 38151 T^{8} + 115245 T^{9} + 740220 T^{10} - 1471990 T^{11} + 17450771 T^{12} - 33416370 T^{13} + 168938315 T^{14} - 313742585 T^{15} + 815730721 T^{16} \))(\( ( 1 - 5 T - 13 T^{2} + 60 T^{3} + 230 T^{4} - 895 T^{5} + 972 T^{6} + 2250 T^{7} - 29091 T^{8} + 29250 T^{9} + 164268 T^{10} - 1966315 T^{11} + 6569030 T^{12} + 22277580 T^{13} - 62748517 T^{14} - 313742585 T^{15} + 815730721 T^{16} )^{2} \))(\( 1 + 7 T - 13 T^{2} - 114 T^{3} + 302 T^{4} + 619 T^{5} - 3203 T^{6} + 11659 T^{7} - 57039 T^{8} - 415732 T^{9} + 983197 T^{10} + 802283 T^{11} - 7060468 T^{12} + 40515806 T^{13} - 160505050 T^{14} - 179149458 T^{15} + 5116354083 T^{16} - 2328942954 T^{17} - 27125353450 T^{18} + 89013225782 T^{19} - 201654026548 T^{20} + 297882061919 T^{21} + 4745704128373 T^{22} - 26086566469444 T^{23} - 46528464595119 T^{24} + 123637858189807 T^{25} - 441560749392347 T^{26} + 1109347283908903 T^{27} + 7036021706989262 T^{28} - 34527762151516842 T^{29} - 51185893014090757 T^{30} + 358301251098635299 T^{31} + 665416609183179841 T^{32} \))(\( 1 - 13 T + 66 T^{2} - 139 T^{3} + 51 T^{4} - 2859 T^{5} + 33752 T^{6} - 143455 T^{7} + 202546 T^{8} + 225355 T^{9} + 2636810 T^{10} - 30800939 T^{11} + 94014139 T^{12} + 15525383 T^{13} - 616206028 T^{14} - 489756397 T^{15} + 8014753706 T^{16} + 3731069987 T^{17} - 135009335494 T^{18} + 334766284429 T^{19} - 343304150125 T^{20} + 4351961697577 T^{21} - 22816577698486 T^{22} + 8197160761439 T^{23} + 228909380597066 T^{24} - 181843121911321 T^{25} - 2974308801804652 T^{26} + 974194759107011 T^{27} + 76690221390664219 T^{28} - 326628538313311247 T^{29} + 363506649892361690 T^{30} + 403872305598208135 T^{31} + 4718933949218036626 T^{32} - 43448948416191654115 T^{33} + \)\(13\!\cdots\!28\)\( T^{34} - \)\(14\!\cdots\!63\)\( T^{35} + 33936247068342171891 T^{36} - \)\(12\!\cdots\!87\)\( T^{37} + \)\(74\!\cdots\!14\)\( T^{38} - \)\(19\!\cdots\!01\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \))
$17$ (\( 1 + 6 T + 19 T^{2} + 132 T^{3} + 829 T^{4} + 2244 T^{5} + 5491 T^{6} + 29478 T^{7} + 83521 T^{8} \))(\( 1 + 7 T + 2 T^{2} - 65 T^{3} - 169 T^{4} - 1105 T^{5} + 578 T^{6} + 34391 T^{7} + 83521 T^{8} \))(\( 1 - 12 T + 37 T^{2} + 180 T^{3} - 1619 T^{4} + 3060 T^{5} + 10693 T^{6} - 58956 T^{7} + 83521 T^{8} \))(\( 1 + 9 T + 44 T^{2} + 243 T^{3} + 1279 T^{4} + 4131 T^{5} + 12716 T^{6} + 44217 T^{7} + 83521 T^{8} \))(\( 1 + 3 T - 8 T^{2} - 75 T^{3} - 89 T^{4} - 1275 T^{5} - 2312 T^{6} + 14739 T^{7} + 83521 T^{8} \))(\( 1 - 4 T - 10 T^{2} + 76 T^{3} - 101 T^{4} - 3812 T^{5} + 15336 T^{6} + 19792 T^{7} - 179499 T^{8} + 336464 T^{9} + 4432104 T^{10} - 18728356 T^{11} - 8435621 T^{12} + 107909132 T^{13} - 241375690 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} \))(\( 1 - 11 T + 39 T^{2} - 139 T^{3} + 1249 T^{4} - 7484 T^{5} + 34098 T^{6} - 128922 T^{7} + 457683 T^{8} - 2191674 T^{9} + 9854322 T^{10} - 36768892 T^{11} + 104317729 T^{12} - 197360123 T^{13} + 941365191 T^{14} - 4513725403 T^{15} + 6975757441 T^{16} \))(\( 1 - T^{2} + 619 T^{4} - 846 T^{6} + 289386 T^{8} - 1318963 T^{10} + 122454076 T^{12} - 275463268 T^{14} + 37914286177 T^{16} - 79608884452 T^{18} + 10227486881596 T^{20} - 31836560420947 T^{22} + 2018686542821226 T^{24} - 1705530839779854 T^{26} + 360643164845222059 T^{28} - 168377826559400929 T^{30} + 48661191875666868481 T^{32} \))(\( 1 - 5 T + 13 T^{2} - 163 T^{3} + 1572 T^{4} - 7139 T^{5} + 24321 T^{6} - 178695 T^{7} + 1262499 T^{8} - 5104293 T^{9} + 17336797 T^{10} - 107294017 T^{11} + 637972982 T^{12} - 2309382396 T^{13} + 7576691718 T^{14} - 42485576390 T^{15} + 218631742615 T^{16} - 722254798630 T^{17} + 2189663906502 T^{18} - 11345995711548 T^{19} + 53284141429622 T^{20} - 152342161095569 T^{21} + 418468133826493 T^{22} - 2094488816223189 T^{23} + 8806886793505059 T^{24} - 21191060590631415 T^{25} + 49030987652820129 T^{26} - 244667067740191987 T^{27} + 915882156925184292 T^{28} - 1614446219363667731 T^{29} + 2188911745272212077 T^{30} - 14312115257549078965 T^{31} + 48661191875666868481 T^{32} \))(\( 1 - T - 34 T^{2} + 41 T^{3} + 811 T^{4} + 3837 T^{5} - 18522 T^{6} - 139403 T^{7} + 389000 T^{8} + 3251903 T^{9} + 2685136 T^{10} - 59936211 T^{11} - 176272043 T^{12} + 1070308079 T^{13} + 4384181682 T^{14} - 6978997953 T^{15} - 68502447326 T^{16} + 33742794795 T^{17} + 1218398812778 T^{18} + 458231992717 T^{19} - 15300717915965 T^{20} + 7789943876189 T^{21} + 352117256892842 T^{22} + 165778350827835 T^{23} - 5721392903114846 T^{24} - 9909179096552721 T^{25} + 105823487857811058 T^{26} + 439188796838039167 T^{27} - 1229631015597521963 T^{28} - 7107707987766132867 T^{29} + 5413217797876026064 T^{30} + \)\(11\!\cdots\!99\)\( T^{31} + \)\(22\!\cdots\!00\)\( T^{32} - \)\(13\!\cdots\!11\)\( T^{33} - \)\(31\!\cdots\!38\)\( T^{34} + \)\(10\!\cdots\!41\)\( T^{35} + \)\(39\!\cdots\!91\)\( T^{36} + \)\(33\!\cdots\!57\)\( T^{37} - \)\(47\!\cdots\!06\)\( T^{38} - \)\(23\!\cdots\!53\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \))
$19$ (\( 1 + 7 T + 15 T^{2} + 107 T^{3} + 824 T^{4} + 2033 T^{5} + 5415 T^{6} + 48013 T^{7} + 130321 T^{8} \))(\( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \))(\( 1 - T - 13 T^{2} - 53 T^{3} + 400 T^{4} - 1007 T^{5} - 4693 T^{6} - 6859 T^{7} + 130321 T^{8} \))(\( 1 - 8 T + 45 T^{2} - 268 T^{3} + 1529 T^{4} - 5092 T^{5} + 16245 T^{6} - 54872 T^{7} + 130321 T^{8} \))(\( 1 + 4 T - 3 T^{2} - 88 T^{3} - 295 T^{4} - 1672 T^{5} - 1083 T^{6} + 27436 T^{7} + 130321 T^{8} \))(\( 1 - 10 T^{2} + 70 T^{3} + 319 T^{4} - 280 T^{5} + 300 T^{6} + 20400 T^{7} + 134681 T^{8} + 387600 T^{9} + 108300 T^{10} - 1920520 T^{11} + 41572399 T^{12} + 173326930 T^{13} - 470458810 T^{14} + 16983563041 T^{16} \))(\( 1 + 9 T + 43 T^{2} + 171 T^{3} + 1023 T^{4} + 1674 T^{5} - 12940 T^{6} - 86580 T^{7} - 253639 T^{8} - 1645020 T^{9} - 4671340 T^{10} + 11481966 T^{11} + 133318383 T^{12} + 423412929 T^{13} + 2022972883 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} \))(\( ( 1 - 9 T + 4 T^{2} + 216 T^{3} - 654 T^{4} - 3357 T^{5} + 21971 T^{6} + 47538 T^{7} - 676108 T^{8} + 903222 T^{9} + 7931531 T^{10} - 23025663 T^{11} - 85229934 T^{12} + 534837384 T^{13} + 188183524 T^{14} - 8044845651 T^{15} + 16983563041 T^{16} )^{2} \))(\( 1 - 19 T + 113 T^{2} + 15 T^{3} - 2596 T^{4} + 2631 T^{5} + 106245 T^{6} - 756527 T^{7} + 1484429 T^{8} + 6206619 T^{9} - 26235105 T^{10} - 133461535 T^{11} + 1444663142 T^{12} - 5677361136 T^{13} + 11109732198 T^{14} - 13150474478 T^{15} + 38443302841 T^{16} - 249859015082 T^{17} + 4010613323478 T^{18} - 38941020031824 T^{19} + 188269945328582 T^{20} - 330463973351965 T^{21} - 1234253627852505 T^{22} + 5547921318840441 T^{23} + 25210893501388589 T^{24} - 244121955937653533 T^{25} + 651395134560067245 T^{26} + 306485871161214189 T^{27} - 5745765529895753956 T^{28} + 630794751933855885 T^{29} + 90287755493465905673 T^{30} - \)\(28\!\cdots\!81\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 - 10 T - 2 T^{2} + 248 T^{3} + 82 T^{4} - 2032 T^{5} - 23994 T^{6} + 41732 T^{7} + 518146 T^{8} - 926242 T^{9} + 540084 T^{10} - 30787074 T^{11} + 50342382 T^{12} + 703180340 T^{13} - 2471883182 T^{14} - 3864155488 T^{15} - 21312045690 T^{16} + 241124991192 T^{17} + 917942443426 T^{18} - 4413260805162 T^{19} - 5762405945050 T^{20} - 83851955298078 T^{21} + 331377222076786 T^{22} + 1653876314585928 T^{23} - 2777407106366490 T^{24} - 9568031539681312 T^{25} - 116291922026273342 T^{26} + 628553033346411260 T^{27} + 854993018331103662 T^{28} - 9934610030411708646 T^{29} + 3311290788778195284 T^{30} - \)\(10\!\cdots\!98\)\( T^{31} + \)\(11\!\cdots\!06\)\( T^{32} + \)\(17\!\cdots\!88\)\( T^{33} - \)\(19\!\cdots\!74\)\( T^{34} - \)\(30\!\cdots\!68\)\( T^{35} + \)\(23\!\cdots\!42\)\( T^{36} + \)\(13\!\cdots\!72\)\( T^{37} - \)\(20\!\cdots\!82\)\( T^{38} - \)\(19\!\cdots\!90\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \))
$23$ (\( ( 1 + 11 T + 75 T^{2} + 253 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 42 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 5 T + 21 T^{2} + 115 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 10 T + 66 T^{2} - 230 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 10 T + 83 T^{2} - 500 T^{3} + 2869 T^{4} - 11500 T^{5} + 43907 T^{6} - 121670 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 8 T + 83 T^{2} - 402 T^{3} + 2555 T^{4} - 9246 T^{5} + 43907 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 123 T^{2} + 7385 T^{4} + 286533 T^{6} + 7800604 T^{8} + 151575957 T^{10} + 2066625785 T^{12} + 18208414347 T^{14} + 78310985281 T^{16} )^{2} \))(\( ( 1 + 16 T + 224 T^{2} + 2234 T^{3} + 19567 T^{4} + 141712 T^{5} + 926670 T^{6} + 5230592 T^{7} + 26816927 T^{8} + 120303616 T^{9} + 490208430 T^{10} + 1724209904 T^{11} + 5475648847 T^{12} + 14378790262 T^{13} + 33160039136 T^{14} + 54477207152 T^{15} + 78310985281 T^{16} )^{2} \))(\( ( 1 + 124 T^{2} - 38 T^{3} + 8330 T^{4} - 3634 T^{5} + 381506 T^{6} - 180342 T^{7} + 12965000 T^{8} - 5936040 T^{9} + 338330610 T^{10} - 136528920 T^{11} + 6858485000 T^{12} - 2194221114 T^{13} + 106761020546 T^{14} - 23389670462 T^{15} + 1233138955370 T^{16} - 129383366986 T^{17} + 9710562174844 T^{18} + 41426511213649 T^{20} )^{2} \))
$29$ (\( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 + 15 T + 106 T^{2} + 675 T^{3} + 4171 T^{4} + 19575 T^{5} + 89146 T^{6} + 365835 T^{7} + 707281 T^{8} \))(\( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 3132 T^{5} - 21025 T^{6} + 48778 T^{7} + 707281 T^{8} \))(\( 1 - 7 T - 10 T^{2} + 113 T^{3} + 59 T^{4} + 3277 T^{5} - 8410 T^{6} - 170723 T^{7} + 707281 T^{8} \))(\( 1 - 3 T - 20 T^{2} + 147 T^{3} + 139 T^{4} + 4263 T^{5} - 16820 T^{6} - 73167 T^{7} + 707281 T^{8} \))(\( 1 - 106 T^{2} + 4095 T^{4} - 50924 T^{6} - 177451 T^{8} - 42827084 T^{10} + 2896315695 T^{12} - 63051272026 T^{14} + 500246412961 T^{16} \))(\( 1 - 9 T - 22 T^{2} + 429 T^{3} - 762 T^{4} - 11754 T^{5} + 81940 T^{6} + 191700 T^{7} - 3727369 T^{8} + 5559300 T^{9} + 68911540 T^{10} - 286668306 T^{11} - 538948122 T^{12} + 8799282921 T^{13} - 13086113062 T^{14} - 155248886781 T^{15} + 500246412961 T^{16} \))(\( 1 - 17 T^{2} + 859 T^{4} - 1960 T^{6} + 281648 T^{8} - 11260925 T^{10} - 142942054 T^{12} + 3210162662 T^{14} - 903405663947 T^{16} + 2699746798742 T^{18} - 101100198895174 T^{20} - 6698260806031925 T^{22} + 140893401717639728 T^{24} - 824586177268393960 T^{26} + \)\(30\!\cdots\!19\)\( T^{28} - \)\(50\!\cdots\!77\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 + 3 T + 75 T^{3} + 2441 T^{4} + 473 T^{5} - 37550 T^{6} + 114715 T^{7} + 3177286 T^{8} - 2623287 T^{9} - 24972736 T^{10} + 263064707 T^{11} + 3471423716 T^{12} - 694532172 T^{13} - 11302302466 T^{14} + 87801320252 T^{15} + 2485262854281 T^{16} + 2546238287308 T^{17} - 9505236373906 T^{18} - 16938945142908 T^{19} + 2455272037276196 T^{20} + 5395759401918343 T^{21} - 14854365761976256 T^{22} - 45251376273007683 T^{23} + 1589425924451203846 T^{24} + 1664187250621812335 T^{25} - 15797556610422547550 T^{26} + 5770841119178857117 T^{27} + \)\(86\!\cdots\!81\)\( T^{28} + \)\(76\!\cdots\!75\)\( T^{29} + \)\(25\!\cdots\!47\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 + 3 T - 4 T^{2} - 59 T^{3} + 609 T^{4} - 7469 T^{5} - 58340 T^{6} - 18675 T^{7} + 564354 T^{8} - 1565053 T^{9} - 1348916 T^{10} + 228651093 T^{11} + 603011151 T^{12} + 2322397715 T^{13} - 7665582012 T^{14} - 36331461243 T^{15} + 746249961816 T^{16} + 576665139899 T^{17} - 30471253369428 T^{18} - 122834837258411 T^{19} + 929104458856239 T^{20} - 3562210280493919 T^{21} - 25626324083688948 T^{22} + 14064286096996711 T^{23} + 527808419243182296 T^{24} - 745200014942898207 T^{25} - 4559666949775701852 T^{26} + 40061073324054233935 T^{27} + \)\(30\!\cdots\!11\)\( T^{28} + \)\(33\!\cdots\!17\)\( T^{29} - \)\(56\!\cdots\!16\)\( T^{30} - \)\(19\!\cdots\!37\)\( T^{31} + \)\(19\!\cdots\!14\)\( T^{32} - \)\(19\!\cdots\!75\)\( T^{33} - \)\(17\!\cdots\!40\)\( T^{34} - \)\(64\!\cdots\!81\)\( T^{35} + \)\(15\!\cdots\!89\)\( T^{36} - \)\(42\!\cdots\!31\)\( T^{37} - \)\(84\!\cdots\!44\)\( T^{38} + \)\(18\!\cdots\!07\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \))
$31$ (\( 1 + 15 T + 69 T^{2} + 95 T^{3} + 36 T^{4} + 2945 T^{5} + 66309 T^{6} + 446865 T^{7} + 923521 T^{8} \))(\( 1 - 13 T + 48 T^{2} + 319 T^{3} - 3835 T^{4} + 9889 T^{5} + 46128 T^{6} - 387283 T^{7} + 923521 T^{8} \))(\( 1 - 17 T + 153 T^{2} - 1129 T^{3} + 7100 T^{4} - 34999 T^{5} + 147033 T^{6} - 506447 T^{7} + 923521 T^{8} \))(\( 1 + 3 T - 22 T^{2} - 159 T^{3} + 205 T^{4} - 4929 T^{5} - 21142 T^{6} + 89373 T^{7} + 923521 T^{8} \))(\( 1 + 3 T - 12 T^{2} + 131 T^{3} + 1365 T^{4} + 4061 T^{5} - 11532 T^{6} + 89373 T^{7} + 923521 T^{8} \))(\( 1 - 24 T + 238 T^{2} - 1194 T^{3} + 1727 T^{4} + 14880 T^{5} - 38008 T^{6} - 810792 T^{7} + 7640665 T^{8} - 25134552 T^{9} - 36525688 T^{10} + 443290080 T^{11} + 1594920767 T^{12} - 34183206294 T^{13} + 211225876078 T^{14} - 660302738664 T^{15} + 852891037441 T^{16} \))(\( 1 + 11 T + 34 T^{2} + 183 T^{3} + 3210 T^{4} + 12836 T^{5} - 31430 T^{6} + 25700 T^{7} + 2517721 T^{8} + 796700 T^{9} - 30204230 T^{10} + 382397276 T^{11} + 2964502410 T^{12} + 5239134633 T^{13} + 30175125154 T^{14} + 302638755221 T^{15} + 852891037441 T^{16} \))(\( ( 1 - 4 T - 20 T^{2} + 75 T^{3} + 855 T^{4} - 3892 T^{5} - 14267 T^{6} + 85685 T^{7} - 118850 T^{8} + 2656235 T^{9} - 13710587 T^{10} - 115946572 T^{11} + 789610455 T^{12} + 2147186325 T^{13} - 17750073620 T^{14} - 110050456444 T^{15} + 852891037441 T^{16} )^{2} \))(\( 1 + 7 T + 13 T^{3} + 77 T^{4} - 8801 T^{5} - 22254 T^{6} + 56119 T^{7} - 947868 T^{8} + 3202093 T^{9} + 57615842 T^{10} - 102279009 T^{11} - 159564504 T^{12} + 5764151836 T^{13} - 36979480986 T^{14} - 181717554252 T^{15} + 841105835367 T^{16} - 5633244181812 T^{17} - 35537281227546 T^{18} + 171719847346276 T^{19} - 147361170298584 T^{20} - 2928161192791359 T^{21} + 51134271858914402 T^{22} + 88097949056534323 T^{23} - 808428121877125788 T^{24} + 1483765156034695849 T^{25} - 18240007898470745454 T^{26} - \)\(22\!\cdots\!31\)\( T^{27} + 60650034351718331597 T^{28} + \)\(31\!\cdots\!83\)\( T^{29} + \)\(16\!\cdots\!57\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 13 T + 65 T^{2} + 146 T^{3} + 295 T^{4} + 2802 T^{5} + 48827 T^{6} + 252198 T^{7} + 448847 T^{8} + 5903225 T^{9} + 65860972 T^{10} + 114046885 T^{11} - 1198096047 T^{12} - 10361131234 T^{13} - 7489384555 T^{14} + 470405899010 T^{15} + 3347900754057 T^{16} + 6862384130018 T^{17} + 25638402034775 T^{18} + 239884734794937 T^{19} + 1493178408224822 T^{20} + 7436426778643047 T^{21} + 24638504355418775 T^{22} + 204437285617366238 T^{23} + 3091856652287474697 T^{24} + 13467321514048040510 T^{25} - 6646856360987046955 T^{26} - \)\(28\!\cdots\!74\)\( T^{27} - \)\(10\!\cdots\!27\)\( T^{28} + \)\(30\!\cdots\!35\)\( T^{29} + \)\(53\!\cdots\!72\)\( T^{30} + \)\(14\!\cdots\!75\)\( T^{31} + \)\(35\!\cdots\!67\)\( T^{32} + \)\(61\!\cdots\!18\)\( T^{33} + \)\(36\!\cdots\!67\)\( T^{34} + \)\(65\!\cdots\!02\)\( T^{35} + \)\(21\!\cdots\!95\)\( T^{36} + \)\(32\!\cdots\!06\)\( T^{37} + \)\(45\!\cdots\!65\)\( T^{38} + \)\(28\!\cdots\!23\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \))
$37$ (\( 1 + 6 T - 21 T^{2} - 248 T^{3} - 471 T^{4} - 9176 T^{5} - 28749 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 22 T + 207 T^{2} - 1220 T^{3} + 6701 T^{4} - 45140 T^{5} + 283383 T^{6} - 1114366 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T + 99 T^{2} + 272 T^{3} + 3969 T^{4} + 10064 T^{5} + 135531 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T - T^{2} - 228 T^{3} - 1331 T^{4} - 8436 T^{5} - 1369 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T + 39 T^{2} + 352 T^{3} + 3309 T^{4} + 13024 T^{5} + 53391 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 6 T + 51 T^{2} - 228 T^{3} + 3191 T^{4} - 8808 T^{5} + 60945 T^{6} - 124230 T^{7} + 3928552 T^{8} - 4596510 T^{9} + 83433705 T^{10} - 446151624 T^{11} + 5980447751 T^{12} - 15810422196 T^{13} + 130852046859 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 - 6 T - 63 T^{2} + 520 T^{3} + 1350 T^{4} - 12954 T^{5} - 20209 T^{6} + 92390 T^{7} + 1366879 T^{8} + 3418430 T^{9} - 27666121 T^{10} - 656158962 T^{11} + 2530117350 T^{12} + 36058857640 T^{13} - 161640763767 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( ( 1 - 12 T + 36 T^{2} + 160 T^{3} - 1779 T^{4} + 12924 T^{5} - 20320 T^{6} - 661816 T^{7} + 6779137 T^{8} - 24487192 T^{9} - 27818080 T^{10} + 654639372 T^{11} - 3334132419 T^{12} + 11095033120 T^{13} + 92366150724 T^{14} - 1139182525596 T^{15} + 3512479453921 T^{16} )^{2} \))(\( 1 - 4 T - 64 T^{2} + 392 T^{3} + 2308 T^{4} - 10820 T^{5} - 47691 T^{6} - 31178 T^{7} + 2748163 T^{8} + 12045860 T^{9} - 124307808 T^{10} - 75116270 T^{11} + 4767089611 T^{12} - 456957494 T^{13} - 122520873497 T^{14} + 217199596514 T^{15} + 4709853143282 T^{16} + 8036385071018 T^{17} - 167731075817393 T^{18} - 23146267943582 T^{19} + 8934293432441371 T^{20} - 5208859396880390 T^{21} - 318939825830501472 T^{22} + 1143536101481319380 T^{23} + 9652866073525897123 T^{24} - 4051947123330910706 T^{25} - \)\(22\!\cdots\!59\)\( T^{26} - \)\(19\!\cdots\!60\)\( T^{27} + \)\(15\!\cdots\!48\)\( T^{28} + \)\(95\!\cdots\!24\)\( T^{29} - \)\(57\!\cdots\!96\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 + 32 T + 496 T^{2} + 5340 T^{3} + 50712 T^{4} + 444144 T^{5} + 3184776 T^{6} + 17020156 T^{7} + 60430936 T^{8} - 66824016 T^{9} - 4566815524 T^{10} - 57414614544 T^{11} - 482483421256 T^{12} - 3283683956236 T^{13} - 19092317847912 T^{14} - 82953054964976 T^{15} - 130679609958264 T^{16} + 1790702451469140 T^{17} + 24993747942875904 T^{18} + 219578098429763712 T^{19} + 1506082500954951158 T^{20} + 8124389641901257344 T^{21} + 34216440933797112576 T^{22} + 90704451274266348420 T^{23} - \)\(24\!\cdots\!04\)\( T^{24} - \)\(57\!\cdots\!32\)\( T^{25} - \)\(48\!\cdots\!08\)\( T^{26} - \)\(31\!\cdots\!88\)\( T^{27} - \)\(16\!\cdots\!76\)\( T^{28} - \)\(74\!\cdots\!88\)\( T^{29} - \)\(21\!\cdots\!76\)\( T^{30} - \)\(11\!\cdots\!08\)\( T^{31} + \)\(39\!\cdots\!16\)\( T^{32} + \)\(41\!\cdots\!32\)\( T^{33} + \)\(28\!\cdots\!64\)\( T^{34} + \)\(14\!\cdots\!92\)\( T^{35} + \)\(62\!\cdots\!92\)\( T^{36} + \)\(24\!\cdots\!80\)\( T^{37} + \)\(83\!\cdots\!84\)\( T^{38} + \)\(19\!\cdots\!36\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \))
$41$ (\( 1 - 6 T + 35 T^{2} - 84 T^{3} - 371 T^{4} - 3444 T^{5} + 58835 T^{6} - 413526 T^{7} + 2825761 T^{8} \))(\( 1 + 13 T + 53 T^{2} + 331 T^{3} + 3380 T^{4} + 13571 T^{5} + 89093 T^{6} + 895973 T^{7} + 2825761 T^{8} \))(\( 1 + 49 T^{2} + 60 T^{3} + 1861 T^{4} + 2460 T^{5} + 82369 T^{6} + 2825761 T^{8} \))(\( 1 - T + 35 T^{2} - 99 T^{3} + 1904 T^{4} - 4059 T^{5} + 58835 T^{6} - 68921 T^{7} + 2825761 T^{8} \))(\( 1 - 15 T + 59 T^{2} - 45 T^{3} + 256 T^{4} - 1845 T^{5} + 99179 T^{6} - 1033815 T^{7} + 2825761 T^{8} \))(\( 1 + 20 T + 170 T^{2} + 1070 T^{3} + 8519 T^{4} + 70060 T^{5} + 465660 T^{6} + 2505360 T^{7} + 13832161 T^{8} + 102719760 T^{9} + 782774460 T^{10} + 4828605260 T^{11} + 24072657959 T^{12} + 123966135070 T^{13} + 807517720970 T^{14} + 3895085477620 T^{15} + 7984925229121 T^{16} \))(\( 1 - 22 T + 160 T^{2} - 414 T^{3} + 2641 T^{4} - 46342 T^{5} + 334288 T^{6} - 1062556 T^{7} + 2274033 T^{8} - 43564796 T^{9} + 561938128 T^{10} - 3193936982 T^{11} + 7462834801 T^{12} - 47964467214 T^{13} + 760016678560 T^{14} - 4284594025382 T^{15} + 7984925229121 T^{16} \))(\( 1 - 155 T^{2} + 10818 T^{4} - 492940 T^{6} + 20344438 T^{8} - 854432905 T^{10} + 36570391811 T^{12} - 1786643351600 T^{14} + 82486521925880 T^{16} - 3003347474039600 T^{18} + 103339186934243171 T^{20} - 4058645365690450105 T^{22} + \)\(16\!\cdots\!98\)\( T^{24} - \)\(66\!\cdots\!40\)\( T^{26} + \)\(24\!\cdots\!58\)\( T^{28} - \)\(58\!\cdots\!55\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 10 T + 32 T^{2} - 192 T^{3} + 5438 T^{4} - 48920 T^{5} + 140400 T^{6} - 769008 T^{7} + 14682039 T^{8} - 96831496 T^{9} + 182717296 T^{10} - 1985070190 T^{11} + 32206741548 T^{12} - 131499578440 T^{13} + 27334729480 T^{14} - 4391234846920 T^{15} + 62202943867805 T^{16} - 180040628723720 T^{17} + 45949680255880 T^{18} - 9063082445663240 T^{19} + 91008554203418028 T^{20} - 229982690931748190 T^{21} + 867926202633652336 T^{22} - 18858347692290955976 T^{23} + \)\(11\!\cdots\!19\)\( T^{24} - \)\(25\!\cdots\!88\)\( T^{25} + \)\(18\!\cdots\!00\)\( T^{26} - \)\(26\!\cdots\!20\)\( T^{27} + \)\(12\!\cdots\!78\)\( T^{28} - \)\(17\!\cdots\!32\)\( T^{29} + \)\(12\!\cdots\!52\)\( T^{30} - \)\(15\!\cdots\!10\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 3 T - 151 T^{2} - 283 T^{3} + 15260 T^{4} + 53807 T^{5} - 908059 T^{6} - 5105899 T^{7} + 41696524 T^{8} + 332390297 T^{9} - 1424132099 T^{10} - 18190782425 T^{11} + 13662607978 T^{12} + 883993384739 T^{13} + 2395186168445 T^{14} - 33386146780263 T^{15} - 238454984830174 T^{16} + 934346443615571 T^{17} + 13814720376087777 T^{18} - 13438189208809197 T^{19} - 619586511510882716 T^{20} - 550965757561177077 T^{21} + 23222544952203553137 T^{22} + 64396091240428768891 T^{23} - \)\(67\!\cdots\!14\)\( T^{24} - \)\(38\!\cdots\!63\)\( T^{25} + \)\(11\!\cdots\!45\)\( T^{26} + \)\(17\!\cdots\!59\)\( T^{27} + \)\(10\!\cdots\!38\)\( T^{28} - \)\(59\!\cdots\!25\)\( T^{29} - \)\(19\!\cdots\!99\)\( T^{30} + \)\(18\!\cdots\!77\)\( T^{31} + \)\(94\!\cdots\!44\)\( T^{32} - \)\(47\!\cdots\!79\)\( T^{33} - \)\(34\!\cdots\!99\)\( T^{34} + \)\(83\!\cdots\!07\)\( T^{35} + \)\(97\!\cdots\!60\)\( T^{36} - \)\(73\!\cdots\!23\)\( T^{37} - \)\(16\!\cdots\!71\)\( T^{38} - \)\(13\!\cdots\!83\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \))
$43$ (\( ( 1 - 6 T + 50 T^{2} - 258 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{4} \))(\( ( 1 - 2 T + 82 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 14 T + 115 T^{2} + 602 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 9 T + 43 T^{2} )^{4} \))(\( ( 1 + 4 T + 69 T^{2} + 392 T^{3} + 4097 T^{4} + 16856 T^{5} + 127581 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 4 T + 45 T^{2} - 172 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 - 8 T + 97 T^{2} - 398 T^{3} + 4048 T^{4} - 17114 T^{5} + 179353 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{4} \))(\( ( 1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 269681724 T^{9} + 3712348240 T^{10} + 7832075556 T^{11} + 93887113062 T^{12} + 137599902648 T^{13} + 1510805768711 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 6 T + 312 T^{2} - 1828 T^{3} + 47319 T^{4} - 259736 T^{5} + 4576717 T^{6} - 22838020 T^{7} + 311402464 T^{8} - 1377461130 T^{9} + 15540959670 T^{10} - 59230828590 T^{11} + 575783155936 T^{12} - 1815782456140 T^{13} + 15646884656317 T^{14} - 38183384951048 T^{15} + 299120578115631 T^{16} - 496884421103596 T^{17} + 3646718486611512 T^{18} - 3015555671621058 T^{19} + 21611482313284249 T^{20} )^{2} \))
$47$ (\( 1 - 10 T + 13 T^{2} - 200 T^{3} + 3549 T^{4} - 9400 T^{5} + 28717 T^{6} - 1038230 T^{7} + 4879681 T^{8} \))(\( 1 - 18 T + 137 T^{2} - 990 T^{3} + 7951 T^{4} - 46530 T^{5} + 302633 T^{6} - 1868814 T^{7} + 4879681 T^{8} \))(\( 1 - 2 T - 43 T^{2} + 180 T^{3} + 1661 T^{4} + 8460 T^{5} - 94987 T^{6} - 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 16 T + 59 T^{2} - 738 T^{3} - 9721 T^{4} - 34686 T^{5} + 130331 T^{6} + 1661168 T^{7} + 4879681 T^{8} \))(\( 1 + 8 T + 67 T^{2} + 610 T^{3} + 6231 T^{4} + 28670 T^{5} + 148003 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( 1 - 22 T + 222 T^{2} - 1554 T^{3} + 9963 T^{4} - 18726 T^{5} - 507516 T^{6} + 6126008 T^{7} - 43788087 T^{8} + 287922376 T^{9} - 1121102844 T^{10} - 1944189498 T^{11} + 48616261803 T^{12} - 356402140878 T^{13} + 2392985803038 T^{14} - 11145708650186 T^{15} + 23811286661761 T^{16} \))(\( 1 + 7 T - 100 T^{2} - 1414 T^{3} + 286 T^{4} + 103523 T^{5} + 579636 T^{6} - 2543800 T^{7} - 42803781 T^{8} - 119558600 T^{9} + 1280415924 T^{10} + 10748068429 T^{11} + 1395588766 T^{12} - 324293839898 T^{13} - 1077921532900 T^{14} + 3546361843241 T^{15} + 23811286661761 T^{16} \))(\( 1 + 150 T^{2} + 11804 T^{4} + 788090 T^{6} + 49820245 T^{8} + 2923059310 T^{10} + 161644493976 T^{12} + 8546050524540 T^{14} + 421499764117049 T^{16} + 18878225608708860 T^{18} + 788773566009301656 T^{20} + 31508285721928162990 T^{22} + \)\(11\!\cdots\!45\)\( T^{24} + \)\(41\!\cdots\!10\)\( T^{26} + \)\(13\!\cdots\!64\)\( T^{28} + \)\(38\!\cdots\!50\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 23 T + 186 T^{2} - 322 T^{3} - 4127 T^{4} + 30138 T^{5} - 70920 T^{6} - 1094344 T^{7} + 18527582 T^{8} - 119631636 T^{9} + 262096090 T^{10} + 2200503287 T^{11} - 30693701772 T^{12} + 88025997822 T^{13} + 1410058928022 T^{14} - 15969608141898 T^{15} + 104939392199145 T^{16} - 750571582669206 T^{17} + 3114820172000598 T^{18} + 9139123171873506 T^{19} - 149775473356494732 T^{20} + 504674441760538009 T^{21} + 2825190190998963610 T^{22} - 60608152736413767468 T^{23} + \)\(44\!\cdots\!02\)\( T^{24} - \)\(12\!\cdots\!48\)\( T^{25} - \)\(37\!\cdots\!80\)\( T^{26} + \)\(74\!\cdots\!14\)\( T^{27} - \)\(47\!\cdots\!07\)\( T^{28} - \)\(17\!\cdots\!94\)\( T^{29} + \)\(47\!\cdots\!34\)\( T^{30} - \)\(27\!\cdots\!89\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 20 T + 141 T^{2} + 502 T^{3} + 4037 T^{4} - 6218 T^{5} - 801001 T^{6} - 7197116 T^{7} - 33989974 T^{8} - 317744608 T^{9} - 1914149671 T^{10} + 9965865086 T^{11} + 157004029439 T^{12} + 867584843046 T^{13} + 9623990203403 T^{14} + 81333165828100 T^{15} + 137676757256356 T^{16} - 1184018410587624 T^{17} - 6791369921638187 T^{18} - 141773649130018030 T^{19} - 1649383233650553737 T^{20} - 6663361509110847410 T^{21} - 15002136156898755083 T^{22} - \)\(12\!\cdots\!52\)\( T^{23} + \)\(67\!\cdots\!36\)\( T^{24} + \)\(18\!\cdots\!00\)\( T^{25} + \)\(10\!\cdots\!87\)\( T^{26} + \)\(43\!\cdots\!98\)\( T^{27} + \)\(37\!\cdots\!79\)\( T^{28} + \)\(11\!\cdots\!62\)\( T^{29} - \)\(10\!\cdots\!79\)\( T^{30} - \)\(78\!\cdots\!24\)\( T^{31} - \)\(39\!\cdots\!34\)\( T^{32} - \)\(39\!\cdots\!32\)\( T^{33} - \)\(20\!\cdots\!69\)\( T^{34} - \)\(75\!\cdots\!74\)\( T^{35} + \)\(22\!\cdots\!77\)\( T^{36} + \)\(13\!\cdots\!74\)\( T^{37} + \)\(17\!\cdots\!49\)\( T^{38} + \)\(11\!\cdots\!60\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 - 14 T + 43 T^{2} + 650 T^{3} - 8799 T^{4} + 34450 T^{5} + 120787 T^{6} - 2084278 T^{7} + 7890481 T^{8} \))(\( 1 - 9 T - 7 T^{2} + 525 T^{3} - 3884 T^{4} + 27825 T^{5} - 19663 T^{6} - 1339893 T^{7} + 7890481 T^{8} \))(\( 1 - 26 T + 323 T^{2} - 2870 T^{3} + 22001 T^{4} - 152110 T^{5} + 907307 T^{6} - 3870802 T^{7} + 7890481 T^{8} \))(\( 1 + 25 T + 257 T^{2} + 1635 T^{3} + 10244 T^{4} + 86655 T^{5} + 721913 T^{6} + 3721925 T^{7} + 7890481 T^{8} \))(\( 1 - 11 T + 43 T^{2} - 565 T^{3} + 6936 T^{4} - 29945 T^{5} + 120787 T^{6} - 1637647 T^{7} + 7890481 T^{8} \))(\( 1 - 20 T + 117 T^{2} - 500 T^{3} + 10595 T^{4} - 96940 T^{5} + 362607 T^{6} - 3666700 T^{7} + 43463984 T^{8} - 194335100 T^{9} + 1018563063 T^{10} - 14432136380 T^{11} + 83599646195 T^{12} - 209097746500 T^{13} + 2593230252093 T^{14} - 23494222796740 T^{15} + 62259690411361 T^{16} \))(\( 1 + 2 T - 173 T^{2} - 964 T^{3} + 9936 T^{4} + 104742 T^{5} + 122655 T^{6} - 3229712 T^{7} - 31796061 T^{8} - 171174736 T^{9} + 344537895 T^{10} + 15593674734 T^{11} + 78399819216 T^{12} - 403140455252 T^{13} - 3834434475317 T^{14} + 2349422279674 T^{15} + 62259690411361 T^{16} \))(\( 1 - 167 T^{2} + 13430 T^{4} - 534760 T^{6} + 4719770 T^{8} + 767763759 T^{10} - 31320486233 T^{12} - 939645984720 T^{14} + 141333507438640 T^{16} - 2639465571078480 T^{18} - 247133701532248073 T^{20} + 17016993216234523911 T^{22} + \)\(29\!\cdots\!70\)\( T^{24} - \)\(93\!\cdots\!40\)\( T^{26} + \)\(65\!\cdots\!30\)\( T^{28} - \)\(23\!\cdots\!23\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 4 T - 22 T^{2} + 440 T^{3} + 660 T^{4} - 49868 T^{5} + 140113 T^{6} + 2285596 T^{7} - 15873165 T^{8} + 76638860 T^{9} + 1323931162 T^{10} - 10376469922 T^{11} - 35381234389 T^{12} + 446229246850 T^{13} - 3132652820265 T^{14} - 9035271835264 T^{15} + 292112805840914 T^{16} - 478869407268992 T^{17} - 8799621772124385 T^{18} + 66433271583287450 T^{19} - 279174957702951109 T^{20} - 4339392954630461546 T^{21} + 29344088384504601898 T^{22} + 90028522586408265820 T^{23} - \)\(98\!\cdots\!65\)\( T^{24} + \)\(75\!\cdots\!68\)\( T^{25} + \)\(24\!\cdots\!37\)\( T^{26} - \)\(46\!\cdots\!96\)\( T^{27} + \)\(32\!\cdots\!60\)\( T^{28} + \)\(11\!\cdots\!20\)\( T^{29} - \)\(30\!\cdots\!18\)\( T^{30} + \)\(29\!\cdots\!28\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 3 T - 126 T^{2} - 1842 T^{3} - 51 T^{4} + 176024 T^{5} + 1687499 T^{6} - 693786 T^{7} - 129427770 T^{8} - 1030515651 T^{9} + 22821416 T^{10} + 58177807579 T^{11} + 439171427598 T^{12} + 484180552566 T^{13} - 15615841075935 T^{14} - 125664083280776 T^{15} - 350599946515409 T^{16} + 1684175373629766 T^{17} + 16682975693935818 T^{18} + 76739832358344117 T^{19} + 165299081934488990 T^{20} + 4067211114992238201 T^{21} + 46862478724265712762 T^{22} + \)\(25\!\cdots\!82\)\( T^{23} - \)\(27\!\cdots\!29\)\( T^{24} - \)\(52\!\cdots\!68\)\( T^{25} - \)\(34\!\cdots\!15\)\( T^{26} + \)\(56\!\cdots\!42\)\( T^{27} + \)\(27\!\cdots\!78\)\( T^{28} + \)\(19\!\cdots\!07\)\( T^{29} + \)\(39\!\cdots\!84\)\( T^{30} - \)\(95\!\cdots\!47\)\( T^{31} - \)\(63\!\cdots\!70\)\( T^{32} - \)\(18\!\cdots\!78\)\( T^{33} + \)\(23\!\cdots\!31\)\( T^{34} + \)\(12\!\cdots\!68\)\( T^{35} - \)\(19\!\cdots\!71\)\( T^{36} - \)\(37\!\cdots\!46\)\( T^{37} - \)\(13\!\cdots\!14\)\( T^{38} + \)\(17\!\cdots\!51\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \))
$59$ (\( 1 - 6 T - 43 T^{2} + 102 T^{3} + 3025 T^{4} + 6018 T^{5} - 149683 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 + 15 T + 31 T^{2} - 885 T^{3} - 10424 T^{4} - 52215 T^{5} + 107911 T^{6} + 3080685 T^{7} + 12117361 T^{8} \))(\( 1 - 14 T + 37 T^{2} + 758 T^{3} - 10095 T^{4} + 44722 T^{5} + 128797 T^{6} - 2875306 T^{7} + 12117361 T^{8} \))(\( 1 + T - 43 T^{2} - 347 T^{3} + 2540 T^{4} - 20473 T^{5} - 149683 T^{6} + 205379 T^{7} + 12117361 T^{8} \))(\( 1 - 9 T + 77 T^{2} - 477 T^{3} + 700 T^{4} - 28143 T^{5} + 268037 T^{6} - 1848411 T^{7} + 12117361 T^{8} \))(\( 1 + 18 T + 22 T^{2} - 1598 T^{3} - 11653 T^{4} - 17790 T^{5} + 84228 T^{6} + 3028056 T^{7} + 40795865 T^{8} + 178655304 T^{9} + 293197668 T^{10} - 3653692410 T^{11} - 141203607733 T^{12} - 1142449029802 T^{13} + 927971740102 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} \))(\( 1 + 25 T + 206 T^{2} - 25 T^{3} - 15220 T^{4} - 157300 T^{5} - 429276 T^{6} + 8492650 T^{7} + 113408229 T^{8} + 501066350 T^{9} - 1494309756 T^{10} - 32306116700 T^{11} - 184426234420 T^{12} - 17873107475 T^{13} + 8689189930046 T^{14} + 62216287120475 T^{15} + 146830437604321 T^{16} \))(\( 1 + 147 T^{2} + 25378 T^{4} + 2282616 T^{6} + 232410162 T^{8} + 15762768105 T^{10} + 1255395853547 T^{12} + 72090796770972 T^{14} + 4968816160696160 T^{16} + 250948063559753532 T^{18} + 15212084755332129467 T^{20} + \)\(66\!\cdots\!05\)\( T^{22} + \)\(34\!\cdots\!02\)\( T^{24} + \)\(11\!\cdots\!16\)\( T^{26} + \)\(45\!\cdots\!18\)\( T^{28} + \)\(91\!\cdots\!67\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 17 T - 50 T^{2} - 1985 T^{3} + 451 T^{4} + 76307 T^{5} - 426220 T^{6} + 1429235 T^{7} + 60743716 T^{8} - 309308083 T^{9} - 1984484046 T^{10} + 33011892753 T^{11} - 101353409674 T^{12} - 2740413120218 T^{13} + 8464635562844 T^{14} + 86256705224328 T^{15} - 317724345351369 T^{16} + 5089145608235352 T^{17} + 29465396394259964 T^{18} - 562823306217252622 T^{19} - 1228135853600750314 T^{20} + 23601004285101705147 T^{21} - 83706596062330791486 T^{22} - \)\(76\!\cdots\!77\)\( T^{23} + \)\(89\!\cdots\!36\)\( T^{24} + \)\(12\!\cdots\!65\)\( T^{25} - \)\(21\!\cdots\!20\)\( T^{26} + \)\(23\!\cdots\!13\)\( T^{27} + \)\(80\!\cdots\!31\)\( T^{28} - \)\(20\!\cdots\!15\)\( T^{29} - \)\(30\!\cdots\!50\)\( T^{30} + \)\(62\!\cdots\!83\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 9 T - 11 T^{2} + 2009 T^{3} - 16566 T^{4} + 39875 T^{5} + 1481949 T^{6} - 16040319 T^{7} + 91259286 T^{8} + 482457503 T^{9} - 9939821477 T^{10} + 75582121635 T^{11} - 12007856396 T^{12} - 4148742606411 T^{13} + 36852203890145 T^{14} - 106193218742209 T^{15} - 1378906768809232 T^{16} + 14879013797272333 T^{17} - 71002723235317991 T^{18} - 551840607836296325 T^{19} + 6375429074335094320 T^{20} - 32558595862341483175 T^{21} - \)\(24\!\cdots\!71\)\( T^{22} + \)\(30\!\cdots\!07\)\( T^{23} - \)\(16\!\cdots\!52\)\( T^{24} - \)\(75\!\cdots\!91\)\( T^{25} + \)\(15\!\cdots\!45\)\( T^{26} - \)\(10\!\cdots\!09\)\( T^{27} - \)\(17\!\cdots\!16\)\( T^{28} + \)\(65\!\cdots\!65\)\( T^{29} - \)\(50\!\cdots\!77\)\( T^{30} + \)\(14\!\cdots\!77\)\( T^{31} + \)\(16\!\cdots\!66\)\( T^{32} - \)\(16\!\cdots\!01\)\( T^{33} + \)\(91\!\cdots\!89\)\( T^{34} + \)\(14\!\cdots\!25\)\( T^{35} - \)\(35\!\cdots\!06\)\( T^{36} + \)\(25\!\cdots\!71\)\( T^{37} - \)\(82\!\cdots\!31\)\( T^{38} - \)\(39\!\cdots\!51\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \))
$61$ (\( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} \))(\( 1 - 3 T + 18 T^{2} + 209 T^{3} + 675 T^{4} + 12749 T^{5} + 66978 T^{6} - 680943 T^{7} + 13845841 T^{8} \))(\( 1 + 20 T + 179 T^{2} + 1600 T^{3} + 15001 T^{4} + 97600 T^{5} + 666059 T^{6} + 4539620 T^{7} + 13845841 T^{8} \))(\( 1 - 19 T + 180 T^{2} - 1781 T^{3} + 17099 T^{4} - 108641 T^{5} + 669780 T^{6} - 4312639 T^{7} + 13845841 T^{8} \))(\( 1 - 15 T + 74 T^{2} - 675 T^{3} + 8491 T^{4} - 41175 T^{5} + 275354 T^{6} - 3404715 T^{7} + 13845841 T^{8} \))(\( 1 + 2 T - 74 T^{2} - 552 T^{3} + 1931 T^{4} + 48386 T^{5} + 199304 T^{6} - 1537696 T^{7} - 12160643 T^{8} - 93799456 T^{9} + 741610184 T^{10} + 10982702666 T^{11} + 26736318971 T^{12} - 466217158152 T^{13} - 3812507702714 T^{14} + 6285485672042 T^{15} + 191707312997281 T^{16} \))(\( 1 - 7 T - 108 T^{2} + 1783 T^{3} + 318 T^{4} - 164342 T^{5} + 1092660 T^{6} + 5279480 T^{7} - 100824709 T^{8} + 322048280 T^{9} + 4065787860 T^{10} - 37302511502 T^{11} + 4402977438 T^{12} + 1505915204683 T^{13} - 5564200430988 T^{14} - 21999199852147 T^{15} + 191707312997281 T^{16} \))(\( ( 1 + 17 T + 63 T^{2} - 314 T^{3} - 258 T^{4} - 10745 T^{5} - 454758 T^{6} - 34468 T^{7} + 32358065 T^{8} - 2102548 T^{9} - 1692154518 T^{10} - 2438910845 T^{11} - 3572226978 T^{12} - 265203238514 T^{13} + 3245783584743 T^{14} + 53426628212357 T^{15} + 191707312997281 T^{16} )^{2} \))(\( 1 + 7 T - 20 T^{2} + 403 T^{3} + 10067 T^{4} + 48739 T^{5} - 116314 T^{6} + 1172749 T^{7} + 26082242 T^{8} + 24039263 T^{9} - 379016678 T^{10} - 5077729829 T^{11} - 113601259714 T^{12} - 664603591174 T^{13} + 819501759744 T^{14} - 46004665392052 T^{15} - 871290620425703 T^{16} - 2806284588915172 T^{17} + 3049366048007424 T^{18} - 150852387728265694 T^{19} - 1572904979399749474 T^{20} - 4288631831050762529 T^{21} - 19527081139622592758 T^{22} + 75549221576474692523 T^{23} + \)\(50\!\cdots\!02\)\( T^{24} + \)\(13\!\cdots\!09\)\( T^{25} - \)\(82\!\cdots\!14\)\( T^{26} + \)\(21\!\cdots\!79\)\( T^{27} + \)\(26\!\cdots\!07\)\( T^{28} + \)\(65\!\cdots\!43\)\( T^{29} - \)\(19\!\cdots\!20\)\( T^{30} + \)\(42\!\cdots\!07\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 + 15 T - 90 T^{2} - 3165 T^{3} - 19585 T^{4} + 92683 T^{5} + 2977530 T^{6} + 30435495 T^{7} + 58060720 T^{8} - 2575904645 T^{9} - 32871179972 T^{10} - 119504505635 T^{11} + 1285823633985 T^{12} + 22572191661785 T^{13} + 150060548339790 T^{14} - 142916662288619 T^{15} - 12943544265704350 T^{16} - 115634960295586275 T^{17} - 240942316112795150 T^{18} + 4864605484258100025 T^{19} + 61052179162441039695 T^{20} + \)\(29\!\cdots\!25\)\( T^{21} - \)\(89\!\cdots\!50\)\( T^{22} - \)\(26\!\cdots\!75\)\( T^{23} - \)\(17\!\cdots\!50\)\( T^{24} - \)\(12\!\cdots\!19\)\( T^{25} + \)\(77\!\cdots\!90\)\( T^{26} + \)\(70\!\cdots\!85\)\( T^{27} + \)\(24\!\cdots\!85\)\( T^{28} - \)\(13\!\cdots\!35\)\( T^{29} - \)\(23\!\cdots\!72\)\( T^{30} - \)\(11\!\cdots\!45\)\( T^{31} + \)\(15\!\cdots\!20\)\( T^{32} + \)\(49\!\cdots\!95\)\( T^{33} + \)\(29\!\cdots\!30\)\( T^{34} + \)\(55\!\cdots\!83\)\( T^{35} - \)\(71\!\cdots\!85\)\( T^{36} - \)\(70\!\cdots\!65\)\( T^{37} - \)\(12\!\cdots\!90\)\( T^{38} + \)\(12\!\cdots\!15\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \))
$67$ (\( ( 1 + 67 T^{2} )^{4} \))(\( ( 1 + 14 T + 178 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 8 T + 67 T^{2} )^{4} \))(\( ( 1 - 22 T + 250 T^{2} - 1474 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 6 T + 98 T^{2} - 402 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 28 T + 541 T^{2} + 6696 T^{3} + 64817 T^{4} + 448632 T^{5} + 2428549 T^{6} + 8421364 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 15 T + 335 T^{2} + 3060 T^{3} + 35713 T^{4} + 205020 T^{5} + 1503815 T^{6} + 4511445 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 18 T + 296 T^{2} + 2706 T^{3} + 26782 T^{4} + 181302 T^{5} + 1328744 T^{6} + 5413734 T^{7} + 20151121 T^{8} )^{4} \))(\( ( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 10079774800 T^{9} + 70488109512 T^{10} + 425631075473 T^{11} + 2398084154605 T^{12} + 10464819704357 T^{13} + 47038358727880 T^{14} + 115153520501137 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 38 T + 939 T^{2} - 16920 T^{3} + 253022 T^{4} - 3231648 T^{5} + 37034650 T^{6} - 384823180 T^{7} + 3720320353 T^{8} - 33446868814 T^{9} + 283134556406 T^{10} - 2240940210538 T^{11} + 16700518064617 T^{12} - 115740574086340 T^{13} + 746289713342650 T^{14} - 4363129101786336 T^{15} + 22887960773164718 T^{16} - 102547240362065160 T^{17} + 381297549225685899 T^{18} - 1033848307059207986 T^{19} + 1822837804551761449 T^{20} )^{2} \))
$71$ (\( 1 - 24 T + 185 T^{2} - 456 T^{3} + 49 T^{4} - 32376 T^{5} + 932585 T^{6} - 8589864 T^{7} + 25411681 T^{8} \))(\( 1 + 13 T - 2 T^{2} - 349 T^{3} + 405 T^{4} - 24779 T^{5} - 10082 T^{6} + 4652843 T^{7} + 25411681 T^{8} \))(\( 1 - 20 T + 169 T^{2} - 1600 T^{3} + 17121 T^{4} - 113600 T^{5} + 851929 T^{6} - 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - T + 70 T^{2} - 119 T^{3} + 5709 T^{4} - 8449 T^{5} + 352870 T^{6} - 357911 T^{7} + 25411681 T^{8} \))(\( 1 - 15 T + 44 T^{2} + 945 T^{3} - 13979 T^{4} + 67095 T^{5} + 221804 T^{6} - 5368665 T^{7} + 25411681 T^{8} \))(\( 1 + 14 T - 77 T^{2} - 2716 T^{3} - 11793 T^{4} + 191640 T^{5} + 2135337 T^{6} - 4898798 T^{7} - 179455320 T^{8} - 347814658 T^{9} + 10764233817 T^{10} + 68590064040 T^{11} - 299679954033 T^{12} - 4900286917316 T^{13} - 9863721861917 T^{14} + 127331682217474 T^{15} + 645753531245761 T^{16} \))(\( 1 - 14 T - 9 T^{2} + 147 T^{3} + 6927 T^{4} - 52059 T^{5} + 728307 T^{6} - 4047848 T^{7} - 14238189 T^{8} - 287397208 T^{9} + 3671395587 T^{10} - 18632488749 T^{11} + 176026714287 T^{12} + 265221714597 T^{13} - 1152902555289 T^{14} - 127331682217474 T^{15} + 645753531245761 T^{16} \))(\( 1 - 163 T^{2} + 12879 T^{4} - 364200 T^{6} - 15289172 T^{8} + 2194637705 T^{10} - 140959687314 T^{12} + 9779417267438 T^{14} - 792564174863307 T^{16} + 49298042445154958 T^{18} - 3582022607883114834 T^{20} + \)\(28\!\cdots\!05\)\( T^{22} - \)\(98\!\cdots\!92\)\( T^{24} - \)\(11\!\cdots\!00\)\( T^{26} + \)\(21\!\cdots\!39\)\( T^{28} - \)\(13\!\cdots\!03\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 - 14 T + 140 T^{2} - 763 T^{3} + 10437 T^{4} - 25566 T^{5} + 32310 T^{6} + 3862236 T^{7} + 10586088 T^{8} + 330657408 T^{9} - 2081383949 T^{10} + 39603959100 T^{11} + 15217861524 T^{12} + 1721196695007 T^{13} + 1593568125167 T^{14} + 22773634065006 T^{15} + 1780879231377332 T^{16} + 1616928018615426 T^{17} + 8033176918966847 T^{18} + 616035230306650377 T^{19} + 386711442550061844 T^{20} + 71454625424023544100 T^{21} - \)\(26\!\cdots\!29\)\( T^{22} + \)\(30\!\cdots\!28\)\( T^{23} + \)\(68\!\cdots\!68\)\( T^{24} + \)\(17\!\cdots\!16\)\( T^{25} + \)\(10\!\cdots\!10\)\( T^{26} - \)\(59\!\cdots\!86\)\( T^{27} + \)\(17\!\cdots\!17\)\( T^{28} - \)\(88\!\cdots\!93\)\( T^{29} + \)\(11\!\cdots\!40\)\( T^{30} - \)\(82\!\cdots\!14\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 + 37 T + 515 T^{2} + 3274 T^{3} + 13215 T^{4} + 211688 T^{5} + 4684917 T^{6} + 60912592 T^{7} + 467488647 T^{8} + 2980461565 T^{9} + 33268020472 T^{10} + 428318804705 T^{11} + 4339324633273 T^{12} + 35873672109364 T^{13} + 288523918504695 T^{14} + 2785853372108440 T^{15} + 26650571131092837 T^{16} + 220452230480408382 T^{17} + 1885088405364310445 T^{18} + 17288624106824488873 T^{19} + \)\(15\!\cdots\!82\)\( T^{20} + \)\(12\!\cdots\!83\)\( T^{21} + \)\(95\!\cdots\!45\)\( T^{22} + \)\(78\!\cdots\!02\)\( T^{23} + \)\(67\!\cdots\!97\)\( T^{24} + \)\(50\!\cdots\!40\)\( T^{25} + \)\(36\!\cdots\!95\)\( T^{26} + \)\(32\!\cdots\!24\)\( T^{27} + \)\(28\!\cdots\!53\)\( T^{28} + \)\(19\!\cdots\!55\)\( T^{29} + \)\(10\!\cdots\!72\)\( T^{30} + \)\(68\!\cdots\!15\)\( T^{31} + \)\(76\!\cdots\!27\)\( T^{32} + \)\(70\!\cdots\!12\)\( T^{33} + \)\(38\!\cdots\!77\)\( T^{34} + \)\(12\!\cdots\!88\)\( T^{35} + \)\(55\!\cdots\!15\)\( T^{36} + \)\(96\!\cdots\!34\)\( T^{37} + \)\(10\!\cdots\!15\)\( T^{38} + \)\(55\!\cdots\!47\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \))
$73$ (\( 1 + 22 T + 111 T^{2} - 1054 T^{3} - 17431 T^{4} - 76942 T^{5} + 591519 T^{6} + 8558374 T^{7} + 28398241 T^{8} \))(\( 1 - 21 T + 233 T^{2} - 2595 T^{3} + 26956 T^{4} - 189435 T^{5} + 1241657 T^{6} - 8169357 T^{7} + 28398241 T^{8} \))(\( 1 + 2 T - 49 T^{2} + 406 T^{3} + 5929 T^{4} + 29638 T^{5} - 261121 T^{6} + 778034 T^{7} + 28398241 T^{8} \))(\( 1 + 11 T + 3 T^{2} - 815 T^{3} - 8464 T^{4} - 59495 T^{5} + 15987 T^{6} + 4279187 T^{7} + 28398241 T^{8} \))(\( 1 + 7 T - 39 T^{2} + 161 T^{3} + 6764 T^{4} + 11753 T^{5} - 207831 T^{6} + 2723119 T^{7} + 28398241 T^{8} \))(\( 1 - 2 T - 146 T^{2} + 446 T^{3} + 13211 T^{4} - 19066 T^{5} - 955660 T^{6} + 409840 T^{7} + 76882177 T^{8} + 29918320 T^{9} - 5092712140 T^{10} - 7416998122 T^{11} + 375169161851 T^{12} + 924589930478 T^{13} - 22094797038194 T^{14} - 22094797038194 T^{15} + 806460091894081 T^{16} \))(\( 1 - 3 T - 2 T^{2} - 345 T^{3} + 220 T^{4} - 100292 T^{5} + 620974 T^{6} + 13580 T^{7} + 31898639 T^{8} + 991340 T^{9} + 3309170446 T^{10} - 39015292964 T^{11} + 6247613020 T^{12} - 715209699585 T^{13} - 302668452578 T^{14} - 33142195557291 T^{15} + 806460091894081 T^{16} \))(\( ( 1 + 21 T + 208 T^{2} + 1428 T^{3} + 11398 T^{4} + 35413 T^{5} - 751679 T^{6} - 12443326 T^{7} - 110271472 T^{8} - 908362798 T^{9} - 4005697391 T^{10} + 13776259021 T^{11} + 323683150918 T^{12} + 2960346234804 T^{13} + 31477519068112 T^{14} + 231995368901037 T^{15} + 806460091894081 T^{16} )^{2} \))(\( 1 + 35 T + 622 T^{2} + 8881 T^{3} + 123217 T^{4} + 1506367 T^{5} + 15880754 T^{6} + 159131205 T^{7} + 1535260024 T^{8} + 13342640169 T^{9} + 108277262748 T^{10} + 880406179959 T^{11} + 6960680213462 T^{12} + 52467783616198 T^{13} + 405906809006112 T^{14} + 3284917386947140 T^{15} + 27500270697009885 T^{16} + 239798969247141220 T^{17} + 2163077385193570848 T^{18} + 20410859779022497366 T^{19} + \)\(19\!\cdots\!42\)\( T^{20} + \)\(18\!\cdots\!87\)\( T^{21} + \)\(16\!\cdots\!72\)\( T^{22} + \)\(14\!\cdots\!93\)\( T^{23} + \)\(12\!\cdots\!44\)\( T^{24} + \)\(93\!\cdots\!65\)\( T^{25} + \)\(68\!\cdots\!46\)\( T^{26} + \)\(47\!\cdots\!59\)\( T^{27} + \)\(28\!\cdots\!57\)\( T^{28} + \)\(14\!\cdots\!73\)\( T^{29} + \)\(75\!\cdots\!98\)\( T^{30} + \)\(31\!\cdots\!95\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 27 T + 117 T^{2} + 3825 T^{3} - 60386 T^{4} + 206645 T^{5} + 5076633 T^{6} - 88912135 T^{7} + 412271510 T^{8} + 5307403509 T^{9} - 94198639093 T^{10} + 498629925791 T^{11} + 2922251864844 T^{12} - 76647599821181 T^{13} + 554955524573989 T^{14} + 735710433225271 T^{15} - 49117193322293732 T^{16} + 443373919368750715 T^{17} - 1123922652740353551 T^{18} - 21964116604269227841 T^{19} + \)\(31\!\cdots\!04\)\( T^{20} - \)\(16\!\cdots\!93\)\( T^{21} - \)\(59\!\cdots\!79\)\( T^{22} + \)\(17\!\cdots\!55\)\( T^{23} - \)\(13\!\cdots\!12\)\( T^{24} + \)\(15\!\cdots\!03\)\( T^{25} + \)\(83\!\cdots\!21\)\( T^{26} - \)\(84\!\cdots\!57\)\( T^{27} + \)\(23\!\cdots\!64\)\( T^{28} + \)\(29\!\cdots\!83\)\( T^{29} - \)\(40\!\cdots\!57\)\( T^{30} + \)\(16\!\cdots\!93\)\( T^{31} + \)\(94\!\cdots\!10\)\( T^{32} - \)\(14\!\cdots\!55\)\( T^{33} + \)\(61\!\cdots\!97\)\( T^{34} + \)\(18\!\cdots\!65\)\( T^{35} - \)\(39\!\cdots\!46\)\( T^{36} + \)\(18\!\cdots\!25\)\( T^{37} + \)\(40\!\cdots\!73\)\( T^{38} - \)\(68\!\cdots\!99\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \))
$79$ (\( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 45820 T^{5} + 131061 T^{6} - 4930390 T^{7} + 38950081 T^{8} \))(\( 1 + 15 T + 11 T^{2} - 1185 T^{3} - 13064 T^{4} - 93615 T^{5} + 68651 T^{6} + 7395585 T^{7} + 38950081 T^{8} \))(\( 1 - 14 T + 117 T^{2} - 532 T^{3} - 1795 T^{4} - 42028 T^{5} + 730197 T^{6} - 6902546 T^{7} + 38950081 T^{8} \))(\( 1 - 13 T + 15 T^{2} - 293 T^{3} + 8624 T^{4} - 23147 T^{5} + 93615 T^{6} - 6409507 T^{7} + 38950081 T^{8} \))(\( 1 + 11 T - 33 T^{2} - 557 T^{3} + 80 T^{4} - 44003 T^{5} - 205953 T^{6} + 5423429 T^{7} + 38950081 T^{8} \))(\( 1 - 20 T + 25 T^{2} + 2320 T^{3} - 19901 T^{4} + 85160 T^{5} - 408605 T^{6} - 11176060 T^{7} + 215428736 T^{8} - 882908740 T^{9} - 2550103805 T^{10} + 41987201240 T^{11} - 775145561981 T^{12} + 7138770845680 T^{13} + 6077186388025 T^{14} - 384078179723180 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 9 T - 75 T^{2} - 768 T^{3} + 10031 T^{4} + 24564 T^{5} - 1553178 T^{6} - 1473567 T^{7} + 120297853 T^{8} - 116411793 T^{9} - 9693383898 T^{10} + 12111009996 T^{11} + 390708262511 T^{12} - 2363179314432 T^{13} - 18231559164075 T^{14} + 172835180875431 T^{15} + 1517108809906561 T^{16} \))(\( ( 1 + 33 T + 372 T^{2} + 792 T^{3} - 10078 T^{4} + 76725 T^{5} + 1586703 T^{6} - 11019954 T^{7} - 280342460 T^{8} - 870576366 T^{9} + 9902613423 T^{10} + 37828417275 T^{11} - 392538916318 T^{12} + 2437028668008 T^{13} + 90428533453812 T^{14} + 633728996543247 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( 1 - 15 T + 94 T^{2} - 2495 T^{3} + 38855 T^{4} - 414785 T^{5} + 5676295 T^{6} - 62322940 T^{7} + 690670255 T^{8} - 8423294280 T^{9} + 83337602737 T^{10} - 856819337855 T^{11} + 8989269459608 T^{12} - 82967566755485 T^{13} + 808287092858990 T^{14} - 7591629044633355 T^{15} + 65763672844350810 T^{16} - 599738694526035045 T^{17} + 5044519746532956590 T^{18} - 40906246145557568915 T^{19} + \)\(35\!\cdots\!48\)\( T^{20} - \)\(26\!\cdots\!45\)\( T^{21} + \)\(20\!\cdots\!77\)\( T^{22} - \)\(16\!\cdots\!20\)\( T^{23} + \)\(10\!\cdots\!55\)\( T^{24} - \)\(74\!\cdots\!60\)\( T^{25} + \)\(53\!\cdots\!95\)\( T^{26} - \)\(31\!\cdots\!15\)\( T^{27} + \)\(22\!\cdots\!55\)\( T^{28} - \)\(11\!\cdots\!05\)\( T^{29} + \)\(34\!\cdots\!14\)\( T^{30} - \)\(43\!\cdots\!85\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 - 5 T - 226 T^{2} + 1188 T^{3} + 28293 T^{4} - 182558 T^{5} - 2094253 T^{6} + 20887148 T^{7} + 66066202 T^{8} - 1685198093 T^{9} + 5203173736 T^{10} + 81171391019 T^{11} - 899452563978 T^{12} + 259180666124 T^{13} + 50685690960293 T^{14} - 453111325450158 T^{15} + 2532348314805923 T^{16} + 41547309775010276 T^{17} - 818375975431692878 T^{18} - 1321261704586397645 T^{19} + 85108949992387444270 T^{20} - \)\(10\!\cdots\!55\)\( T^{21} - \)\(51\!\cdots\!98\)\( T^{22} + \)\(20\!\cdots\!64\)\( T^{23} + \)\(98\!\cdots\!63\)\( T^{24} - \)\(13\!\cdots\!42\)\( T^{25} + \)\(12\!\cdots\!53\)\( T^{26} + \)\(49\!\cdots\!16\)\( T^{27} - \)\(13\!\cdots\!58\)\( T^{28} + \)\(97\!\cdots\!61\)\( T^{29} + \)\(49\!\cdots\!36\)\( T^{30} - \)\(12\!\cdots\!47\)\( T^{31} + \)\(39\!\cdots\!82\)\( T^{32} + \)\(97\!\cdots\!72\)\( T^{33} - \)\(77\!\cdots\!93\)\( T^{34} - \)\(53\!\cdots\!42\)\( T^{35} + \)\(65\!\cdots\!53\)\( T^{36} + \)\(21\!\cdots\!92\)\( T^{37} - \)\(32\!\cdots\!86\)\( T^{38} - \)\(56\!\cdots\!95\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \))
$83$ (\( 1 - 2 T + 121 T^{2} - 736 T^{3} + 7989 T^{4} - 61088 T^{5} + 833569 T^{6} - 1143574 T^{7} + 47458321 T^{8} \))(\( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 64740 T^{5} - 323783 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 14 T + 193 T^{2} - 2140 T^{3} + 26421 T^{4} - 177620 T^{5} + 1329577 T^{6} - 8005018 T^{7} + 47458321 T^{8} \))(\( 1 - 14 T - 7 T^{2} + 640 T^{3} - 2059 T^{4} + 53120 T^{5} - 48223 T^{6} - 8005018 T^{7} + 47458321 T^{8} \))(\( 1 - 14 T + 193 T^{2} - 2140 T^{3} + 26421 T^{4} - 177620 T^{5} + 1329577 T^{6} - 8005018 T^{7} + 47458321 T^{8} \))(\( 1 - 8 T - 10 T^{2} + 98 T^{3} + 5199 T^{4} - 59504 T^{5} + 80364 T^{6} + 1623536 T^{7} + 2309081 T^{8} + 134753488 T^{9} + 553627596 T^{10} - 34023613648 T^{11} + 246735810879 T^{12} + 386025983014 T^{13} - 3269403733690 T^{14} - 217088407917016 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 23 T + 98 T^{2} - 1745 T^{3} - 15470 T^{4} + 98632 T^{5} + 1750974 T^{6} - 1504700 T^{7} - 137136651 T^{8} - 124890100 T^{9} + 12062459886 T^{10} + 56396495384 T^{11} - 734180225870 T^{12} - 6873625922035 T^{13} + 32040156590162 T^{14} + 624129172761421 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 384 T^{2} + 59474 T^{4} - 2689144 T^{6} - 502020269 T^{8} + 90973066528 T^{10} - 4332158192844 T^{12} - 395418317256672 T^{14} + 67591318619685797 T^{16} - 2724036787581213408 T^{18} - \)\(20\!\cdots\!24\)\( T^{20} + \)\(29\!\cdots\!32\)\( T^{22} - \)\(11\!\cdots\!29\)\( T^{24} - \)\(41\!\cdots\!56\)\( T^{26} + \)\(63\!\cdots\!14\)\( T^{28} - \)\(28\!\cdots\!36\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 5 T - 318 T^{2} - 3741 T^{3} + 48517 T^{4} + 994133 T^{5} - 3255786 T^{6} - 170309105 T^{7} - 263285806 T^{8} + 21750909411 T^{9} + 119250152558 T^{10} - 2097202277489 T^{11} - 20999477370338 T^{12} + 141328121959992 T^{13} + 2545394345691772 T^{14} - 4513257769614610 T^{15} - 237031046695965905 T^{16} - 374600394878012630 T^{17} + 17535221647470617308 T^{18} + 80809582871137945704 T^{19} - \)\(99\!\cdots\!98\)\( T^{20} - \)\(82\!\cdots\!27\)\( T^{21} + \)\(38\!\cdots\!02\)\( T^{22} + \)\(59\!\cdots\!97\)\( T^{23} - \)\(59\!\cdots\!46\)\( T^{24} - \)\(31\!\cdots\!15\)\( T^{25} - \)\(50\!\cdots\!14\)\( T^{26} + \)\(12\!\cdots\!11\)\( T^{27} + \)\(51\!\cdots\!37\)\( T^{28} - \)\(33\!\cdots\!83\)\( T^{29} - \)\(23\!\cdots\!22\)\( T^{30} + \)\(30\!\cdots\!35\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 42 T + 661 T^{2} - 3786 T^{3} - 19549 T^{4} + 561934 T^{5} - 5968953 T^{6} + 20177190 T^{7} + 432561126 T^{8} - 6143013970 T^{9} + 28916165545 T^{10} - 66576463426 T^{11} + 144701828969 T^{12} + 5596879899702 T^{13} - 23882904399413 T^{14} + 248862974645262 T^{15} - 21903508692713044 T^{16} + 235757216137931078 T^{17} + 483849794965596541 T^{18} - 23802972835548797594 T^{19} + \)\(24\!\cdots\!45\)\( T^{20} - \)\(19\!\cdots\!02\)\( T^{21} + \)\(33\!\cdots\!49\)\( T^{22} + \)\(13\!\cdots\!86\)\( T^{23} - \)\(10\!\cdots\!24\)\( T^{24} + \)\(98\!\cdots\!66\)\( T^{25} - \)\(78\!\cdots\!97\)\( T^{26} + \)\(15\!\cdots\!54\)\( T^{27} + \)\(32\!\cdots\!29\)\( T^{28} - \)\(12\!\cdots\!78\)\( T^{29} + \)\(44\!\cdots\!05\)\( T^{30} - \)\(79\!\cdots\!90\)\( T^{31} + \)\(46\!\cdots\!86\)\( T^{32} + \)\(17\!\cdots\!70\)\( T^{33} - \)\(43\!\cdots\!37\)\( T^{34} + \)\(34\!\cdots\!38\)\( T^{35} - \)\(99\!\cdots\!69\)\( T^{36} - \)\(15\!\cdots\!78\)\( T^{37} + \)\(23\!\cdots\!49\)\( T^{38} - \)\(12\!\cdots\!74\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \))
$89$ (\( ( 1 + 9 T + 197 T^{2} + 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 15 T + 233 T^{2} - 1335 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 9 T + 167 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 13 T + 159 T^{2} - 1157 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 21 T + 227 T^{2} + 1869 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 16 T + 308 T^{2} - 3096 T^{3} + 38678 T^{4} - 275544 T^{5} + 2439668 T^{6} - 11279504 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 17 T + 312 T^{2} - 3419 T^{3} + 38939 T^{4} - 304291 T^{5} + 2471352 T^{6} - 11984473 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 188 T^{2} + 20573 T^{4} + 1044676 T^{6} + 71173405 T^{8} + 8274878596 T^{10} + 1290796124093 T^{12} + 93432482700668 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 37 T + 1032 T^{2} + 20901 T^{3} + 357823 T^{4} + 5152639 T^{5} + 65606528 T^{6} + 731972999 T^{7} + 7339853952 T^{8} + 65145596911 T^{9} + 519669308288 T^{10} + 3632450763191 T^{11} + 22450616901343 T^{12} + 116712426543549 T^{13} + 512884692271752 T^{14} + 1636559391134573 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 9 T + 640 T^{2} - 4050 T^{3} + 178355 T^{4} - 698876 T^{5} + 29313265 T^{6} - 51732294 T^{7} + 3369419368 T^{8} - 1036172651 T^{9} + 317703552166 T^{10} - 92219365939 T^{11} + 26689170813928 T^{12} - 36469663568886 T^{13} + 1839179937126865 T^{14} - 3902565131479324 T^{15} + 88639098149349155 T^{16} - 179136906326892450 T^{17} + 2519416835649331840 T^{18} - 3153207633367366881 T^{19} + 31181719929966183601 T^{20} )^{2} \))
$97$ (\( 1 - 6 T - 61 T^{2} + 948 T^{3} + 229 T^{4} + 91956 T^{5} - 573949 T^{6} - 5476038 T^{7} + 88529281 T^{8} \))(\( 1 - 17 T + 192 T^{2} - 1615 T^{3} + 8831 T^{4} - 156655 T^{5} + 1806528 T^{6} - 15515441 T^{7} + 88529281 T^{8} \))(\( 1 + 2 T + 27 T^{2} + 760 T^{3} + 10181 T^{4} + 73720 T^{5} + 254043 T^{6} + 1825346 T^{7} + 88529281 T^{8} \))(\( 1 + 3 T + 182 T^{2} - 345 T^{3} + 16591 T^{4} - 33465 T^{5} + 1712438 T^{6} + 2738019 T^{7} + 88529281 T^{8} \))(\( 1 - 33 T + 452 T^{2} - 4035 T^{3} + 36031 T^{4} - 391395 T^{5} + 4252868 T^{6} - 30118209 T^{7} + 88529281 T^{8} \))(\( 1 - 4 T + 26 T^{2} - 1532 T^{3} + 8091 T^{4} + 62828 T^{5} + 1010040 T^{6} - 5954240 T^{7} - 108387563 T^{8} - 577561280 T^{9} + 9503466360 T^{10} + 57341419244 T^{11} + 716290412571 T^{12} - 13155805273724 T^{13} + 21657272128154 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 30 T + 361 T^{2} - 2160 T^{3} + 10062 T^{4} - 83370 T^{5} - 577417 T^{6} + 32754150 T^{7} - 458148745 T^{8} + 3177152550 T^{9} - 5432916553 T^{10} - 76089548010 T^{11} + 890781625422 T^{12} - 18548654955120 T^{13} + 300702893779369 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16} \))(\( ( 1 - 27 T + 301 T^{2} - 2340 T^{3} + 20706 T^{4} - 231291 T^{5} + 2774900 T^{6} - 31398696 T^{7} + 326142677 T^{8} - 3045673512 T^{9} + 26109034100 T^{10} - 211093050843 T^{11} + 1833087292386 T^{12} - 20094376201380 T^{13} + 250724573483629 T^{14} - 2181553680909051 T^{15} + 7837433594376961 T^{16} )^{2} \))(\( 1 - 20 T + 123 T^{2} + 28 T^{3} + 4997 T^{4} - 239296 T^{5} + 2683386 T^{6} - 13918670 T^{7} + 12108314 T^{8} + 81926108 T^{9} + 9118270907 T^{10} - 123570735168 T^{11} + 816960697702 T^{12} + 3552108922056 T^{13} - 86512314985382 T^{14} - 43450963654390 T^{15} + 5424229096525225 T^{16} - 4214743474475830 T^{17} - 813994371697459238 T^{18} + 3241913906219615688 T^{19} + 72324943172816412262 T^{20} - \)\(10\!\cdots\!76\)\( T^{21} + \)\(75\!\cdots\!03\)\( T^{22} + \)\(66\!\cdots\!04\)\( T^{23} + \)\(94\!\cdots\!54\)\( T^{24} - \)\(10\!\cdots\!90\)\( T^{25} + \)\(19\!\cdots\!14\)\( T^{26} - \)\(17\!\cdots\!88\)\( T^{27} + \)\(34\!\cdots\!77\)\( T^{28} + \)\(18\!\cdots\!56\)\( T^{29} + \)\(80\!\cdots\!87\)\( T^{30} - \)\(12\!\cdots\!60\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 27 T - 146 T^{2} - 10565 T^{3} - 42323 T^{4} + 1720741 T^{5} + 15722390 T^{6} - 118678851 T^{7} - 2079209414 T^{8} - 3980410181 T^{9} + 86169204810 T^{10} + 1330688850015 T^{11} + 13895090921091 T^{12} - 18497220903723 T^{13} - 2497628718197930 T^{14} - 21619068182342995 T^{15} + 116376235825961792 T^{16} + 3142408121720289719 T^{17} + 13315354121489765710 T^{18} - \)\(14\!\cdots\!49\)\( T^{19} - \)\(24\!\cdots\!21\)\( T^{20} - \)\(14\!\cdots\!53\)\( T^{21} + \)\(12\!\cdots\!90\)\( T^{22} + \)\(28\!\cdots\!87\)\( T^{23} + \)\(10\!\cdots\!52\)\( T^{24} - \)\(18\!\cdots\!15\)\( T^{25} - \)\(20\!\cdots\!70\)\( T^{26} - \)\(14\!\cdots\!99\)\( T^{27} + \)\(10\!\cdots\!51\)\( T^{28} + \)\(10\!\cdots\!55\)\( T^{29} + \)\(63\!\cdots\!90\)\( T^{30} - \)\(28\!\cdots\!93\)\( T^{31} - \)\(14\!\cdots\!74\)\( T^{32} - \)\(79\!\cdots\!27\)\( T^{33} + \)\(10\!\cdots\!10\)\( T^{34} + \)\(10\!\cdots\!13\)\( T^{35} - \)\(25\!\cdots\!83\)\( T^{36} - \)\(62\!\cdots\!05\)\( T^{37} - \)\(84\!\cdots\!94\)\( T^{38} + \)\(15\!\cdots\!91\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \))
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